Unit 2 notes
Transcript of Unit 2 notes
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October 02, 2012
May 1112:48 PM
Unit 2 PolynomialsAt a birthday party, Crusty the Clown took the age of the birthday girl and multiplied it by 6,
subtracted 2, multiplied it by 7, and then added 3. The resulting number was 493. How old is
the birthday girl?
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POLYNOMIALS
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A polynomial is made up of terms that are only added, subtracted or multiplied.
What is a polynomial???
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A polynomial with 3 terms is called a trinomial.
Monomial = polynomial with 1 termBinomial = polynomial with 2 terms
There is also a quadrinomial (4 terms)& a quintinomial (5 terms) but we rarely
use these words.
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The degree of a polynomial with only one variable is the largest exponent of that variable.
If the polynomial has more than one variable, you need to look at each term. You find the degree of the polynomial by adding the exponents of each term and naming the largest power.
The degree of this trinomial is 4.
The degree of this trinomial is 3.
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Find the degree of each polynomial. 1. 3x2
2. 4x 2y3 ‐ 1
3. 3x2 +4x2y3 ‐ 5z4
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May 147:55 AM
The standard form of a polynomial is to put the terms with the highest degree first.
Arrange the following in standard form:
1. 4x2 ‐ 2x + 3 ‐ 5x3
2. ‐1 + x3 ‐ 9x2 + 8x
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Quiz Time!!!
Just kidding...but here are a few questions...
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Adding & Subtracting Polynomials
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Remember, to add polynomials simply combine like terms. To subtract polynomials, distribute
the -1 and then combine like terms.
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Homework:p. 113 #2- 58 eoe
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10. 22.
30. 46.
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Answers to p. 113 #2 58 eoe
2. 4 6. 3
10. 1 14. 10t + 8
18. 5a b 5ab 22. 5x 6
26. 2x 2x 2 30. 3x x x 6
34. 11x + 8 38. x + 4x + x 5
42. 9x 9x 7 46. n n + 1
50. 3x + 7 54. 3x 2x + 10
58. 4x 4
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Maya takes a number, multiplies it by 4, subtracts 5, multiplies by 4 again
and then adds 8. The resulting number is 228. What number did
Maya start with?
WARM UP
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Multiplying Monomials
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Examples:1. 5(-7x) =
2. -5y(-7x2) =
3. 5x3y2(7xy4) =
4. 5xy(-9x2y3)(4x3y) =
5. 3x(-2x2y3)(-5x3y) =
To multiply 2 polynomials:• Multiply coefficients• Add exponents of like bases
Remember that whenyou multiply the samebases, you ADD the exponent. Don't forgetto multiply coefficients!
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Jul 129:14 AM
Can you put together the "old" stuff you learned yesterday with the "new" stuff from today???
Simplify:
1. (x - 2y) - 4(-x + y) + 2(-4x + 5y)
2. 2m4n(7m5n3 + 6mn2) - 6mn2(2m4n - 7m5n3)
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Power to a Power • Distribute the outside exponent
to all inside exponentsOR
• Rewrite what is inside the parenthesis the same # as the
outside exponent and follow your rules for multiplying polynomials.
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Examples:1. (3xy2)3 =
2. (-x4y5)4 =
3. (-5a2b2c)3 =
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Homework:p. 120 #6- 78Multiples of 6!
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Answers to p. 120 #6 78 (multiples of 6)
6. 12.
18. 24.
30. 36.
42. 48.
54. 60.
66. 72.
76.
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May 141:01 PM
Aidan spent $14.95 on ingredients for cookies that will be sold at a bake sale. If each cookie will sell for $0.40, how many cookies must Aidan sell to earn a
profit?
Warm Up
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Multiplying Polynomials
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Multiplying 2 Binomials
See next page for the process and answer...
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May 141:14 PM
A square has a side represented by x -2. Find the area of the square in terms of x.
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A rectangle has a length represented by 5x + 3 and a width represented by 2x - 7. Find the area and write your answer in standard form.
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May 141:14 PM
A checker board is made up of 16 congruent squares. Each square has a length represented by x - 7. Find the area of 1 square in terms of x. Then find the area of the entire checker board in terms of x.
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Homework:p. 127
#2- 58 eoe
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Quiz Today
CR #2 due tomorrow!
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October 02, 2012
May 1112:56 PM
A hot air balloon begins ascending at the rate of 18 feet per minute. A second hot air balloon begins ascending 8 minutes later at a rate of 27 feet per minute. How many minutes will it take for the second hot air balloon to catch the first hot air
balloon?
Warm Up
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Factoring
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The numbers that are multiplied together are called
factors of the final number.
What is a factor?
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Greatest Common Factor
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Try factoring these examples:1. 28y2 - 4y =
2. 7x3 + 10x2 =
3. 12x3y4 - 39 x4y3 =
4. 5x + 7x2 + 9x4 =
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Factoring the Difference of 2 Perfect Squares
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Try factoring these examples:
1. x2 - 100 =
2. 9 - 25y2 =
3. 16x2 - 1 =
4. x6 - y10 =
5. 49x2 - 64y2 =
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Homework:p. 135 #28- 40 even
p. 142 #1- 13 odd
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Warm UpThe formula 180﴾n 2﴿ is used to find the
sum of the interior angles of any polygon. If n represents the number of sides the polygon has, find the total sum of the interior angles of an octagon.
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Factoring Trinomials
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= (x )(x )
= (x 3)(x 4)
= (x + 3)(x + 4) + for the letter t for the word "BOTH." Both signs are the same as the middle
sign.
Positive. Therefore both
signs are positive.
Here's a little trick (Bigger/ Both)
to help you figure out the signs.
What 2 #s multiply to give you 12 but ADD to
give you 7?
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= (x )(x )
= (x 9)(x 5)
= (x - 9)(x + 5) - for the letter i for the word "BIGGER." The bigger # gets the
same sign as the middle sign.
Negative. Therefore the
bigger # (9) gets the - and the smaller # (5) gets the +.
Here's a little trick (Bigger/ Both)
to help you figure out the signs.
What 2 #s multiply to give you 45 but SUBTRACT to
give you 4?
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Homework:Worksheet
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Warm UpAkira scored 18 points in her last
basketball game, p of which came in the 4th quarter of the game. Write an
expression that can be used to find the number of points she scored in the first 3
quarters.
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Factoring by Grouping
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Let's start out simple...Factor the following:
1.
2.
Factoring by
grouping will pop
up if you go on to
take Calculus!
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KEY POINT!!!
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The Arc Method of Factoring
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Factor:5x2 8x + 3
Wait there are no 2 #'s that multiply to give me 3 and add to give me 8!!!(3)(1) = 33 + 1 = 4
What am I supposed to do????
We will use the Arc Method of Factoring!
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The Arc Method of FactoringSteps:1. Take the coefficient in front of the x2 and multiply it by the last term.2. Replace the last term with this new number.3. Factor this new trinomial. 4. "Un arc" the factors by dividing the #'s by the original # in from of the x2.5. Reduce all fractions.6. If the fraction is not divisible, move the denominator in from of your x term.
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= 5x2 - 8x + 3= x2 - 8x + 15= (x - 5)(x - 3)= (x - )(x - )= (x - 1)(5x - 3)
Multiply to check this answer!!!
5(3) = 15
"Un- Arcing"
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Homework:
Worksheet
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Unit 2- Quiz #3 today!
CR #2 due tomorrow
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Warm UpThe Drama Club is designing a set for an
upcoming show. The structure is shown in the picture. Layer one has 1 triangle, but layer 2 consists of 3 triangles. How many triangles will
make up the 5th layer?
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Factoring Completely
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Types of Factoring:1. GCFExample: 3x2 + 9x3 + 12x4 = 3x2﴾1 + 3x + 4x2﴿
2. Difference between 2 perfect squaresExample: y2 9 = ﴾y + 3﴿﴾y 3﴿
3. TrinomialsExample: x2 2x 3 = ﴾x 3﴿﴾x + 1﴿
How would you factor x 3 + 7x2 + 12x???
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Factoring completely is a three step process:
1. Factor a GCF from the expression, if possible. 2. Factor a Trinomial, if possible. 3. Factor a Difference Between Two Squares as many times as possible.
x3 + 7x2 + 12x = x﴾x2 + 7x + 12﴿ x﴾x + 3﴿﴾x + 4﴿
To factor x 3 + 7x2 + 12x you need to FACTOR COMPLETELY
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Factor each of the following COMPLETELY:
1. 2x2 8
2. n3 49n
3. 25x2 100
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Factor each of the following COMPLETELY:
1. 18x3 + 39x2 15x
2. 3x2 + 15x 18
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Homework:Worksheet
Quiz tomorrow!!!
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Warm Up
What is the next number in the
sequence?
1, 2, 4, 8, 16, ...
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Solving Quadratic Equations
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What is a quadratic? A quadratic equation???
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It must get harder than this, right???
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What do we call the highest power of a
polynomial????
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Solve the equation below for x. How many values will you end up with? Why?
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Here's an interesting
fact...
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Remember the Arc Method!!!3x2 + 13x - 10x2 + 13x - 30(x + 15)(x - 2)(x + )(x - )(x +5)(3x -2)
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October 02, 2012
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What do I do if the quadratic doesn't have 4
parts?
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Here's a super dooper hard one!
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Homework:Worksheet
Review tomorrow!Test in 2 days =)
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