Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide...
Transcript of Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide...
![Page 1: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/1.jpg)
Polynomial Division Primer
John Banks and Liz Bailey
I We can divide polynomials using the same techniques of longdivision as we use with natural numbers – but we getpolynomials as our quotient and remainder.
I We first explain how this works before showing the quicktabular way to do the calculations.
I This slide presentation is best viewed on screen in full screenmode.
I Just view one slide at a time and try to understand what ishappening before moving on.
1 / 63
![Page 2: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/2.jpg)
What are we trying to do?
I In the following examples, we divide the polynomial p(x) bythe polynomial d(x), obtaining an answer in the form
p(x)
d(x)= q(x) +
r(x)
d(x).
where r(x) has (strictly) smaller degree than d(x).
I This is achieved by repeatedly subtracting appropriatemultiples of d(x).
I We choose these multiples so that the “leading” term isremoved from p(x) by each subtraction.
I We can do this most efficiently in a tabular format, but firstlets see why it works.
2 / 63
![Page 3: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/3.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
3 / 63
![Page 4: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/4.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
4 / 63
![Page 5: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/5.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
5 / 63
![Page 6: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/6.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
6 / 63
![Page 7: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/7.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
7 / 63
![Page 8: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/8.jpg)
First Example: Step 1
p(x) = x4 + 3x3 − 3x + 5, d(x) = x + 2.
I To remove the leading term x4 of p(x), subtract x3d(x) . . .
p(x)− x3d(x) = (x4 + 3x3 − 3x + 5)− (x4 + 2x3)
= x3 − 3x + 5
⇒ p(x)− x3d(x)
d(x)=
x3 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 +
x3 − 3x + 5
d(x)
sop(x)
d(x)is x3 plus a fraction
p1(x)
d(x)with numerator
p1(x) = x3 − 3x + 5 of smaller degree than p(x).
I We can do exactly the same thing to this new fraction!
8 / 63
![Page 9: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/9.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
9 / 63
![Page 10: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/10.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
10 / 63
![Page 11: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/11.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
11 / 63
![Page 12: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/12.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
12 / 63
![Page 13: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/13.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
13 / 63
![Page 14: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/14.jpg)
First Example: Step 2
I To remove the leading term x3 of p1(x), subtract x2d(x) . . .
p1(x)− x2d(x) = (x3 − 3x + 5)− (x3 + 2x2)
= −2x2 − 3x + 5
⇒ p1(x)− x2d(x)
d(x)=−2x2 − 3x + 5
d(x)
⇒ p1(x)
d(x)= x2 +
−2x2 − 3x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 +
−2x2 − 3x + 5
d(x)
sop(x)
d(x)is x3 + x2 plus a fraction
p2(x)
d(x)with numerator
p2(x) = −2x2 − 3x + 5 of even smaller degree.
I So what do we do next . . .
14 / 63
![Page 15: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/15.jpg)
First Example: Step 3
I . . . remove the leading term −2x2 of p2(x), by subtracting(−2x)d(x) of course!
p1(x) + 2xd(x) = (−2x2 − 3x + 5) + (2x2 + 4x)
= x + 5
⇒ p2(x) + 2xd(x)
d(x)=
x + 5
d(x)
⇒ p2(x)
d(x)= −2x +
x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x +
x + 5
d(x)
. . . sop(x)
d(x)is x3 + x2 − 2x plus a fraction in which the
numerator p3(x) = x + 5 has even smaller degree.
I This is so much fun. Lets do it again!
15 / 63
![Page 16: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/16.jpg)
First Example: Step 3
I . . . remove the leading term −2x2 of p2(x), by subtracting(−2x)d(x) of course!
p1(x) + 2xd(x) = (−2x2 − 3x + 5) + (2x2 + 4x)
= x + 5
⇒ p2(x) + 2xd(x)
d(x)=
x + 5
d(x)
⇒ p2(x)
d(x)= −2x +
x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x +
x + 5
d(x)
. . . sop(x)
d(x)is x3 + x2 − 2x plus a fraction in which the
numerator p3(x) = x + 5 has even smaller degree.
I This is so much fun. Lets do it again!
16 / 63
![Page 17: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/17.jpg)
First Example: Step 3
I . . . remove the leading term −2x2 of p2(x), by subtracting(−2x)d(x) of course!
p1(x) + 2xd(x) = (−2x2 − 3x + 5) + (2x2 + 4x)
= x + 5
⇒ p2(x) + 2xd(x)
d(x)=
x + 5
d(x)
⇒ p2(x)
d(x)= −2x +
x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x +
x + 5
d(x)
. . . sop(x)
d(x)is x3 + x2 − 2x plus a fraction in which the
numerator p3(x) = x + 5 has even smaller degree.
I This is so much fun. Lets do it again!
17 / 63
![Page 18: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/18.jpg)
First Example: Step 3
I . . . remove the leading term −2x2 of p2(x), by subtracting(−2x)d(x) of course!
p1(x) + 2xd(x) = (−2x2 − 3x + 5) + (2x2 + 4x)
= x + 5
⇒ p2(x) + 2xd(x)
d(x)=
x + 5
d(x)
⇒ p2(x)
d(x)= −2x +
x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x +
x + 5
d(x)
. . . sop(x)
d(x)is x3 + x2 − 2x plus a fraction in which the
numerator p3(x) = x + 5 has even smaller degree.
I This is so much fun. Lets do it again!
18 / 63
![Page 19: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/19.jpg)
First Example: Step 3
I . . . remove the leading term −2x2 of p2(x), by subtracting(−2x)d(x) of course!
p1(x) + 2xd(x) = (−2x2 − 3x + 5) + (2x2 + 4x)
= x + 5
⇒ p2(x) + 2xd(x)
d(x)=
x + 5
d(x)
⇒ p2(x)
d(x)= −2x +
x + 5
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x +
x + 5
d(x)
. . . sop(x)
d(x)is x3 + x2 − 2x plus a fraction in which the
numerator p3(x) = x + 5 has even smaller degree.
I This is so much fun. Lets do it again!
19 / 63
![Page 20: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/20.jpg)
First Example: Step 4
I To remove the leading term x of p3(x), subtract 1× d(x):
p1(x)− d(x) = (x + 5)− (x + 2) = 3
⇒ p3(x) + 2xd(x)
d(x)=
3
d(x)
⇒ p3(x)
d(x)= −2x +
3
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x + 1 +
3
d(x)
. . . expressingp(x)
d(x)as a polynomial x3 + x2 − 2x + 1 plus a
fraction in which the numerator p4(x) = 3 has even smallerdegree.
I Can we do this again?
20 / 63
![Page 21: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/21.jpg)
First Example: Step 4
I To remove the leading term x of p3(x), subtract 1× d(x):
p1(x)− d(x) = (x + 5)− (x + 2) = 3
⇒ p3(x) + 2xd(x)
d(x)=
3
d(x)
⇒ p3(x)
d(x)= −2x +
3
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x + 1 +
3
d(x)
. . . expressingp(x)
d(x)as a polynomial x3 + x2 − 2x + 1 plus a
fraction in which the numerator p4(x) = 3 has even smallerdegree.
I Can we do this again?
21 / 63
![Page 22: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/22.jpg)
First Example: Step 4
I To remove the leading term x of p3(x), subtract 1× d(x):
p1(x)− d(x) = (x + 5)− (x + 2) = 3
⇒ p3(x) + 2xd(x)
d(x)=
3
d(x)
⇒ p3(x)
d(x)= −2x +
3
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x + 1 +
3
d(x)
. . . expressingp(x)
d(x)as a polynomial x3 + x2 − 2x + 1 plus a
fraction in which the numerator p4(x) = 3 has even smallerdegree.
I Can we do this again?
22 / 63
![Page 23: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/23.jpg)
First Example: Step 4
I To remove the leading term x of p3(x), subtract 1× d(x):
p1(x)− d(x) = (x + 5)− (x + 2) = 3
⇒ p3(x) + 2xd(x)
d(x)=
3
d(x)
⇒ p3(x)
d(x)= −2x +
3
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x + 1 +
3
d(x)
. . . expressingp(x)
d(x)as a polynomial x3 + x2 − 2x + 1 plus a
fraction in which the numerator p4(x) = 3 has even smallerdegree.
I Can we do this again?
23 / 63
![Page 24: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/24.jpg)
First Example: Step 4
I To remove the leading term x of p3(x), subtract 1× d(x):
p1(x)− d(x) = (x + 5)− (x + 2) = 3
⇒ p3(x) + 2xd(x)
d(x)=
3
d(x)
⇒ p3(x)
d(x)= −2x +
3
d(x)
⇒ p(x)
d(x)= x3 + x2 − 2x + 1 +
3
d(x)
. . . expressingp(x)
d(x)as a polynomial x3 + x2 − 2x + 1 plus a
fraction in which the numerator p4(x) = 3 has even smallerdegree.
I Can we do this again?
24 / 63
![Page 25: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/25.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . . but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
25 / 63
![Page 26: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/26.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . .
but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
26 / 63
![Page 27: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/27.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . . but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
27 / 63
![Page 28: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/28.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . . but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
28 / 63
![Page 29: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/29.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . . but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
29 / 63
![Page 30: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/30.jpg)
First Example: The end!
I No! The leading term 3 of p4(x) = 3 is now too small (indegree) to express as a multiple of d(x).
I So we cannot repeat the procedure . . . but we now have ananswer in which all possible polynomial terms have beenextracted:
p(x)
d(x)= x3 + x2 − 2x + 1 +
3
x + 2
I Our job is done!
I That was fun, but involved a lot of writing.
I We can carry out the division much more efficiently in atabular format.
30 / 63
![Page 31: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/31.jpg)
Same Example in Tabular Format
x + 2) x4 + 3x3 + 0x2 − 3x + 5
31 / 63
![Page 32: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/32.jpg)
Same Example in Tabular Format
x + 2) x4 + 3x3 + 0x2 − 3x + 5Easier to maintain alignmentif we add any “missing powers”
32 / 63
![Page 33: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/33.jpg)
Same Example in Tabular Format
x + 2) x4 + 3x3 + 0x2 − 3x + 5Need a product of x + 2that removes x4 . . .
33 / 63
![Page 34: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/34.jpg)
Same Example in Tabular Format
x3
x + 2) x4 + 3x3 + 0x2 − 3x + 5. . . so multiply by x3.
34 / 63
![Page 35: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/35.jpg)
Same Example in Tabular Format
x3
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3(x + 2) = x4 + 2x3.
35 / 63
![Page 36: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/36.jpg)
Same Example in Tabular Format
x3
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
Subtracting gives
x3 + 0x2 − 3x + 5.
36 / 63
![Page 37: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/37.jpg)
Same Example in Tabular Format
x3
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5 Need a product of x + 2that removes x3 . . .
37 / 63
![Page 38: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/38.jpg)
Same Example in Tabular Format
x3 + x2
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5 . . . so multiply by x2.
38 / 63
![Page 39: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/39.jpg)
Same Example in Tabular Format
x3 + x2
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2 x2(x + 2) = x3 + 2x2.
39 / 63
![Page 40: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/40.jpg)
Same Example in Tabular Format
x3 + x2
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5 Subtracting gives −2x2 − 3x + 5.
40 / 63
![Page 41: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/41.jpg)
Same Example in Tabular Format
x3 + x2
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5 Need a product of x + 2that removes −2x2 . . .
41 / 63
![Page 42: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/42.jpg)
Same Example in Tabular Format
x3 + x2 − 2x
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5 . . . so multiply by −2x .
42 / 63
![Page 43: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/43.jpg)
Same Example in Tabular Format
x3 + x2 − 2x
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x −2x(x + 2) = −2x2 − 4x .
43 / 63
![Page 44: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/44.jpg)
Same Example in Tabular Format
x3 + x2 − 2x
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5 Subtracting gives x + 5.
44 / 63
![Page 45: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/45.jpg)
Same Example in Tabular Format
x3 + x2 − 2x
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5 Need a product of x + 2that removes x . . .
45 / 63
![Page 46: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/46.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5 . . . so multiply by 1.
46 / 63
![Page 47: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/47.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5
x + 2 1× (x + 2) = x + 2.
47 / 63
![Page 48: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/48.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5
x + 2
3 Subtracting gives 3and we stop.
48 / 63
![Page 49: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/49.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5
x + 2
3
Answer is:
polynomial plus remainder
p(x)
d(x)= x3 + x2 − 2x + 1 + 3
x+2
49 / 63
![Page 50: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/50.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5
x + 2
3
Answer is:
polynomial plus remainder
p(x)
d(x)= x3 + x2 − 2x + 1 + 3
x+2
50 / 63
![Page 51: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/51.jpg)
Same Example in Tabular Format
x3 + x2 − 2x + 1
x + 2) x4 + 3x3 + 0x2 − 3x + 5
x4 + 2x3
x3 + 0x2 − 3x + 5
x3 + 2x2
− 2x2 − 3x + 5
−2x2 − 4x
x + 5
x + 2
3
Answer is:
polynomial plus remainder
p(x)
d(x)= x3 + x2 − 2x + 1 + 3
x+2
When we subtract, we canleave out unnecessary terms.
51 / 63
![Page 52: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/52.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
2x − 1) 8x3 − 4x2 − 2x + 1
52 / 63
![Page 53: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/53.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
2x − 1) 8x3 − 4x2 − 2x + 1
53 / 63
![Page 54: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/54.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
2x − 1) 8x3 − 4x2 − 2x + 1
54 / 63
![Page 55: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/55.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2
2x − 1) 8x3 − 4x2 − 2x + 1
55 / 63
![Page 56: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/56.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
56 / 63
![Page 57: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/57.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
57 / 63
![Page 58: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/58.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
58 / 63
![Page 59: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/59.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2 − 1
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
59 / 63
![Page 60: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/60.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2 − 1
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
60 / 63
![Page 61: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/61.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2 − 1
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
−2x + 1
61 / 63
![Page 62: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/62.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2 − 1
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
−2x + 1
0
62 / 63
![Page 63: Polynomial Division Primer · Polynomial Division Primer John Banks and Liz Bailey I We can divide polynomials using the same techniques of long division as we use with natural numbers](https://reader035.fdocuments.net/reader035/viewer/2022071002/5fbf17ef30fec378685222a4/html5/thumbnails/63.jpg)
Another example
I When we subtract, we sometimes get lucky!
p(x) = 8x3 − 4x2 − 2x + 1, d(x) = 2x − 1.
4x2 − 1
2x − 1) 8x3 − 4x2 − 2x + 1
8x3 − 4x2
0 − 2x + 1
−2x + 1
0
Answer:
p(x)
d(x)= 4x2 − 1 +
0
2x − 1
= 4x2 − 1
63 / 63