What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin...
-
Upload
alexia-shelton -
Category
Documents
-
view
223 -
download
2
Transcript of What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin...
![Page 1: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/1.jpg)
What is a rational expression?It is a ratio of two polynomial expressions, like this:
We will begin by reducing fractions
Now we will reduce Polynomials
Simplify rational expressions means that we could not reduce or factor anything else out of the expression.
Simplify, Multiply, and DivideRational Expressions
![Page 2: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/2.jpg)
Now let’s reduce Polynomials
When dividing polynomials, they are called ____rational expressions____
There are two steps for reducing/simplifying rational expressions.
Step 1. ___Factor the numerator and denominator___
Step 2. _____Reduce/Cancel like terms.
![Page 3: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/3.jpg)
Simplify
Look for common factors.
Simplify.Answer:
![Page 4: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/4.jpg)
Under what conditions is this expression undefined?
A rational expression is undefined if the denominator equals zero. To find out when this expression is undefined, completely factor the denominator.
Answer: The values that would make the denominator equal to 0 are –7,
3, and –3. So the expression is undefined at y = –7, y = 3, and y = –3. These values are called excluded values.
![Page 5: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/5.jpg)
a. Simplify
b. Under what conditions is this expression undefined?
Answer:
Answer: undefined for x = –5, x = 4, x = –4
![Page 6: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/6.jpg)
Multiple-Choice Test Item
For what values of p is undefined?
A 5 B –3, 5 C 3, –5 D 5, 1, –3
Read the Test ItemYou want to determine which values of p make the denominator equal to 0.
![Page 7: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/7.jpg)
Solve the Test ItemLook at the possible answers. Notice that the p term and the constant term are both negative, so there will be one positive solution and one negative solution. Therefore, you can eliminate choices A and D. Factor the denominator.
Factor the denominator.
Solve each equation.
Answer: B
Zero Product Propertyor
![Page 8: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/8.jpg)
Multiple-Choice Test Item
For what values of p is undefined?
A –5, –3, –2 B –5 C 5 D –5, –3
Answer: D
![Page 9: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/9.jpg)
Simplify
Simplify.Answer: or –a
![Page 10: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/10.jpg)
Simplify
Answer: –x
![Page 11: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/11.jpg)
Simplify
Answer: Simplify.
![Page 12: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/12.jpg)
Simplify
Answer: Simplify.
![Page 13: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/13.jpg)
Simplify each expression.
a.
b.
Answer:
Answer:
![Page 14: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/14.jpg)
Simplify
Answer: Simplify.
![Page 15: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/15.jpg)
Simplify
Answer:
![Page 16: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/16.jpg)
Simplify
Answer: Simplify.
![Page 17: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/17.jpg)
Simplify
Simplify.Answer:
![Page 18: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/18.jpg)
Stopped here after day 1
![Page 19: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/19.jpg)
Answer: 1
Simplify each expression.
a.
b.
Answer:
![Page 20: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/20.jpg)
Simplify
Express as adivision expression.
Multiply by thereciprocal of divisor.
![Page 21: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/21.jpg)
Factor.
1 1 –1
1 1 1
Simplify.Answer:
![Page 22: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/22.jpg)
Simplify
Answer:
![Page 23: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/23.jpg)
![Page 24: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/24.jpg)
Example 1 LCM of Monomials
Example 2 LCM of Polynomials
Example 3 Monomial Denominators
Example 4 Polynomial Denominators
Example 5 Simplify Complex Fractions
Example 6 Use a Complex Fraction to Solve a Problem
![Page 25: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/25.jpg)
Find the LCM of 15a2bc3, 16b5c2, and 20a3c6.
Factor the firstmonomial.
Factor the secondmonomial.
Factor the thirdmonomial.
![Page 26: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/26.jpg)
Use each factor the greatest number of times it appears as a factor and simplify.
Answer:
![Page 27: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/27.jpg)
Find the LCM of 6x2zy3, 9x3y2z2, and 4x2z.
Answer: 36x3y3z2
![Page 28: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/28.jpg)
Find the LCM of x3 – x2 – 2x and x2 – 4x + 4.
Factor the first polynomial.
Factor the second polynomial.
Answer: Use each factor the greatest number of times it appears as a factor.
![Page 29: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/29.jpg)
Find the LCM of x3 + 2x2 – 3x and x2 + 6x + 9.
Answer:
![Page 30: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/30.jpg)
Simplify
The LCD is 42a2b2. Find equivalent fractions that have this denominator.
Simplify each numerator and denominator.
Add the numerators.Answer:
![Page 31: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/31.jpg)
Simplify
Answer:
![Page 32: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/32.jpg)
Simplify
Factor the denominators.
The LCD is6(x – 5).
Subtract the numerators.
![Page 33: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/33.jpg)
DistributiveProperty
Combine liketerms.
Simplify.
1
1
Simplify.Answer:
![Page 34: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/34.jpg)
Simplify
Answer:
![Page 35: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/35.jpg)
Simplify
The LCD of the numerator is ab. The LCD of the denominator is b.
![Page 36: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/36.jpg)
Simplify the numerator and denominator.
Write as a divisionexpression.
Multiply by the reciprocal of the divisor.
1
1
![Page 37: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/37.jpg)
Simplify.Answer:
![Page 38: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/38.jpg)
Simplify
Answer:
![Page 39: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/39.jpg)
Coordinate Geometry Find the slope of the line that
passes through and
Definition of slope
![Page 40: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/40.jpg)
The LCD of the numerator is 3k. The LCD of the denominator is 2k.
Write as a division expression.
Simplify.
Answer: The slope is
![Page 41: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/41.jpg)
Coordinate Geometry Find the slope of the line that
passes through and
Answer:
![Page 42: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/42.jpg)
![Page 43: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/43.jpg)
![Page 44: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/44.jpg)
Example 1 Vertical Asymptotes and Point Discontinuity
Example 2 Graph with a Vertical Asymptote
Example 3 Graph with Point Discontinuity
Example 4 Use Graphs of Rational Functions
![Page 45: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/45.jpg)
Determine the equations of any vertical asymptotes and the values of
x for any holes in the graph of
First factor the numerator and denominator of the rational expression.
![Page 46: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/46.jpg)
Answer: The function is undefined for x = –2 and –3.
Since x = –3 is a vertical
asymptote and x = –2 is a hole in the graph.
1
1
![Page 47: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/47.jpg)
Determine the equations of any vertical asymptotes and the values of
x for any holes in the graph of
Answer: vertical asymptote: x = –5; hole: x = –3
![Page 48: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/48.jpg)
Answer:Graph
The function is undefined for
x = –1. Since is in its
simplest form, x = –1 is a vertical asymptote. Draw the vertical asymptote.
![Page 49: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/49.jpg)
Make a table of values.
x f (x)
–4 1.33
–3 1.5
–2 2
0 0
1 0.5
2 0.67
3 0.75
Answer:
Plot the points and draw the graph.
![Page 50: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/50.jpg)
As |x| increases, it appears that the
y values of the function get closer and closer to 1. The line with
the equation f (x) = 1 is a
horizontal asymptote of
the function.
Answer:
![Page 51: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/51.jpg)
Graph
Answer:
![Page 52: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/52.jpg)
Graph
Notice that or Therefore,
the graph of is the graph of
with a hole at
![Page 53: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/53.jpg)
Answer:
![Page 54: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/54.jpg)
Graph
Answer:
![Page 55: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/55.jpg)
Transportation A train travels at one velocity V1 for a given amount of
time t1 and then another velocity V2 for a different amount of time t2. The average
velocity is given by
Let t1 be the independent variable and let V be the dependent
variable. Draw the graph if V1 = 50 miles per hour, V2 = 30 miles
per hour, and t2 = 1 hour.
![Page 56: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/56.jpg)
Answer:
The function is
The vertical asymptote is Graph the vertical
asymptote and the function. Notice that the horizontal
asymptote is
![Page 57: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/57.jpg)
What is the V-intercept of the graph?
Answer: The V-intercept
is 30.
![Page 58: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/58.jpg)
What values of t1 and V are meaningful in the context of the problem?
Answer: In the problem context, time and velocity are positive values. Therefore, positive values
of t1 and V values between 30 and 60 are meaningful.
![Page 59: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/59.jpg)
Transportation A train travels at one velocity V1 for a given amount of
time t1 and then another velocity V2 for a different amount of time t2. The average
velocity is given by
a. Let t1 be the independent variable and let V be the
dependent variable. Draw the graph if V1 = 60 miles
per hour, V2 = 30 miles per hour, and t2 = 1 hour.
![Page 60: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/60.jpg)
Answer:
![Page 61: What is a rational expression? It is a ratio of two polynomial expressions, like this: We will begin by reducing fractions Now we will reduce Polynomials.](https://reader036.fdocuments.net/reader036/viewer/2022062517/56649ee75503460f94bf83f7/html5/thumbnails/61.jpg)
b. What is the V-intercept of the graph?
c. What values of t1 and V are
meaningful in the context of
the problem?
Answer: The V-intercept is 30.
Answer: t1 is positive and V is between 30
and 60.