Polynomials and Factoring

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Polynomials and Factoring
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Aim 9-1 How do we add and subtract polynomials?

• Monomial is an expression that is a number, a
  variable or a product of a number.
• Ex. 7x2y5 Degree 7 (Add the exponents 2 5 7)
• Degree of a monomial is the sum of the
  exponents of its variable .
• Ex. 3x4 The degree is 4.
• - 4 The degree is 0.

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What is a polynomial?

• Polynomial is a monomial or the sum or difference
  of two or more monomials.

• Standard form of a polynomial means that the
  degree of the monomial terms decrease from left
  to right.
• Ex 2x3 x2 x 3

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Classifying Polynomials

• Name the polynomial based on its degree and the
  number of its terms.
• -2x 5

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Classifying Polynomials

• Name the polynomial based on its degree and the
  number of its terms.
• -2x 5
• Answer Linear binomial

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Classifying Polynomials

• Write each in standard form. Then name the
  polynomial based on its degree and the number of
  its terms.
• 3x4 4 2x2 5x4
• Answer 8x4 2x2 4
• fourth degree trinomial

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Practice

• Write each in standard form. Then name the
  polynomial based on its degree and the number of
  its terms.
• 6x2 7 9x4
• 3y 4 y3
• 8 7v 11v

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Adding Polynomials

• (4x2 6x 7) ( 2x2 9x 1)
• Hint Combine like terms.
• Answer 6x2 3x 8

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Practice

• Simplify each sum.
• (12m2 4) (8m2 5)
• (t2 6) ( 3t2 11)
• (2p3 6p2 10p) (9p3 11p2 3p)

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Subtracting Polynomials

• (2x3 5x2 3x ) ( x3 8x2 11)
• The signs of the second polynomial change to
  their opposite.
• (2x3 5x2 3x ) ( - x3 8x2 - 11)
• Now you can add the expressions by combining like
  terms
• .
• Answer x3 13x2 3x - 11

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Summary

• Write a polynomial and identify the following
• The degree
• Number of terms
• Explain how to add and subtract polynomials.

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Aim 9-2 How do we multiply a polynomial by a
monomial?

• -4y2( 5y4 3y2 2)
• Multiply the 4y2 with each term inside the
  parenthesis.
• Answer -20 y6 12 y4 8 y2

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Finding the greatest common factor

• 4x3 12x2 - 8x
• (Whats the GCF of 4, 12, and 8?
• Whats the GCF of x3, x2, and x?)
• The GCF is 4x.

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Practice

• Find the GCF of the terms of each polynomial.
• 5v5 10v3
• 3t2 18
• 4b3 2b2 6b

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Factoring Out a Monomial

• 3x3 12x2 15x
• Think What is the GCF of each term?
• GCF is 3x.
• Answer 3x (x2 4x 5)

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Practice

• Factor Out a Monomial. Use the GCF to factor each
  monomial.
• 8x2 12x
• 5d3 10d
• 6m3 12m2 24m

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Summary

• Explain how do you find the GCF of a polynomial.

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Aim 9-3 How do we multiply binomials using FOIL?

• Strategy 1 Using the distributive property
• ( x 4) (2x 3) Distribute x 4.
• 2x (x 4) 3 (x 4) Then simplify.
• 2x2 8x 3x 12 Combine like terms.
• Answer 2x2 11x 12

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Using the Distributive Property

• Simplify each product.
• 1. (6h 7 ) ( 2h 3 )
• 2. (5m 2) ( 8m 1)

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• Using the FOIL method
• F- First terms
• O- outer terms
• I- Inner terms
• L- last terms
• Example (3x 5) (2x 7)
• F O I L
• 3x 2x 3x 7 (-5) 2x (-5)7
• Then simplify.
• 6x2 21x -10x -35 6x2 11x - 35

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Simplify using FOIL.

• (3x 4) (2x 5)
• (3x 4) (2x 5)

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Applying Multiplication of Polynomials

• Suppose you have a rectangle with dimensions 2x
  5 and 3x 1 and inside you have a smaller
  rectangle with the dimensions x 2 and x. What
  is the area of the unshaded region?

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Multiplying a Trinomial and a Binomial

• (4x2 x 6) (2x 3)
• You may use the distributive property.
• OR you may use the vertical method.
• 4x2 x 6
• 2x 3
• -12x2-3x 18
• 8x32x2 -12x 0
• 8x3-10x2 -15x 18

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Simplify

• (6n 8 ) (2n2 n 7)
• The binomial goes on the bottom.

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Summary

• What do the letters in FOIL represent?

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Aim 9-4How do we find the square of a binomial?

• Investigation
• Exploring Special Products
• Complete on page 474

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Square of a Binomial

• (a b)2 (a b ) (a b) a2 2ab b2
• (a b)2
• Example (x 7)2 x2 2(7x) 49
• x2 14x 49

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Square of a Binomial

• (a b)2 (a b) (a b) a2 - 2ab b2
• (a b)2
• Example (4k 3)2 16k2 2(4k)(3) 9
• 16k2 -24k 9

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(a b)2 a2 2ab b2(a b)2 a2 - 2ab
b2

• Find (y 11)2

• Find (3w 6)2

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Practice Find each square.

• (t 6)2
• (5y 1)2
• (7m 2p)2
• (9c 8)2

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Difference of Squares

• (a b ) (a b) a2 ab ab b2
• a2 - b2
• Example (d 11) (d 11)
• d2 11d 11d 121
• d 2 121

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Practice Find each product.

• (c2 8)(c2 8)
• (p4 8)(p4 8)
• (9v3 w)(9v3 w)

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Summary

• Describe in words how to square a binomial. Give
  an example to support your statement.
• Describe in words the difference of two squares.
  Give an example to support your statement.

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Aim 9-5How do we factor trinomials of the type
x2 bx c?

• Example 1 x2 7x 12

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x2bx c

• Example 1 x2 7x 12
• The factors 3 and 4 will work.
• When 3 and 4 are multiplied 12, which is the c
  term.
• When 3 and 4 are added 7 which is the b term.
• Now we can factor the trinomial.
• Answer ( x 3) ( x 4) How can we check
  our answer?
• 
  Does the order matter?
• Note All terms are positive so all the factors
  are positive too.

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Guided Practice

• g2 7g 10
• What are the factors of 10? (List in pairs)
• 5 and 2, 1 and 10
• Which of the above when added 7?
• 5 and 2
• Now we are ready to factor the trinomial.
• Answer (g 5) (g 2)

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Practice

• Factor each expression.
• Check your answer.
• 1. v2 21v 20 2. a2 13a 30

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How do we factor x2 - bx c?

• d2 17d 42
• Note the b term is negative so the factors of 42
  must be negative because when you add them they
  must -17 and when you multiply them they must
  42.
• List the factors of 42.
• -6 and -7, -1 and -42, -3 and -14
• Answer (d 3) (d - 14)

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Practice

• Factor each expression.
• k2 -10k 25
• x2 11x 18
• q2 -15q 36

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How do we factor x2 bx - c?

• m2 6m 27
• Note the b term is positive and the c term is
  negative. This means one factor will be positive
  and one will be negative.
• When you add them they must equal 6 and when you
  multiply them they must equal -27.
• List the factors of 27.
• 27 and 1, -3 and 9
• Answer (m 3) (m 9)

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Practice

• Factor the expression.
• m2 8m - 20

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How do we factor x2 bx c?

• p2 3p 18
• Note the b term is negative and the c term is
  negative. One factor will be positive and one
  will be negative. Since the b term is negative
  that means the larger of the two factors will be
  negative.
• List the factor of 18.
• -9 and 2, -18 and 1, -6 and 3
• Answer (p 6) (p 3)

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Practice

• Factor each expression.
• 1. p2 - 3p - 40 2. y2 - y - 56

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Summary

• How do you determine what numbers are used in the
  binomial factors when factoring expressions of
  the type
• x2 bx c?

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Aim 9-6 How do we factor ax2 bx c?

• 6n2 23n 7
• Note a is now greater than 1.
• Product of a Product of c
• (x ) (x )
• Sum of products is b

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• 6n2 23n
  7
• F O I
  L
• 1 6 1 7 1 6 13 1 7
• 1 6 1 1 6 7 43 7 1
• 2 3 2 7 3 1 17 1 7
• 2 1 3 7 23? 7
  1

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• 6n2 23n
  7
• 2 1 3 7 23?
• (2n 7 ) ( 3n 1 ) Check your answer!!

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Guided Practice

• Factor each expression.
• 2y2 5y 2
• ( 2y 2 ) ( y 1 ) Does this work?
• What else can we try?
• Answer ( 2y 1 ) ( y 2 )
• Try 6n2 23n 7 and 2y2 5y 2

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How do we factor ax2 - bx c?

• 7x2 - 26x 8
• What are the factors of 7?
• 1 and 7
• What are the factors of 8?
• 2 and 4, 1 and 8
• So does (7x - 2)(x 4) work?
• When you add the cross products it 26.

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• 7x2 - 26x 8
• What should we try?
• (7x 2) (x 4)
• Does the sum of the cross products -26?
• Yes!

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Practice

• Factor each expression and check your answer.
• 5d2 14d 3
• 2n2 n 3
• 20p2 31p - 9

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Factoring Out a Monomial First

• 20 x2 80 x 35
• 1. Factor out the GCF.
• 5(4x2 16x 7)
• 2. Now you can factor the trinomial.
• (2x 1) (2x 7)
• 3.Remember to include the GCF in your final
  answer.
• 5 (2x 1) (2x 7)

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Practice

• Factor each expression.
• 2v2 -12v 10
• 4y2 14y 6
• 18k2 12 k - 6

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Summary

• What is the first thing you should look at when
  factoring a trinomial?
• Ticket Out
• Complete on the Post-It Note
• Factor the expression.
• 18x2 33x - 30

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Aim 9-7 How do we factor perfect square
trinomials?

• Activity
• Factor each trinomial.
• x2 6x 9 b. x2 10 x 9
• m2 15m 36 d. m2 12m 36
• e. k2 26 k 25 f. k2 10 k 25
• Which binomials have pairs of binomial factors
  that are identical?
• Describe the relationship between the middle and
  last terms of the trinomials that have identical
  pairs of factors.

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• Solution
• a.( x 3) ( x 3) b. (x 9) (x 1)
• c. (m 3) (m 12) d. (m 6) (m 6)
• e. (k 25) (k 1) f. (k 5) (k 5)
• x2 6x 9, m2 12m 36 k2 10 k 25
• The middle term is the sum of the identical
  factors.

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Perfect Square Trinomials

• (a b)2 a2 2ab b2(a b)2 a2 - 2ab
  b2
• Examples x2 10x 25 (x 5)(x 5)
• ( x
  5)2
• x2 -10x 25 ?

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Factor x2 8x 16

• Factor the first term x x
• Factor the last term -4, -4 - 8
• (x 4) ( x 4) (x 4)2
• Try
• x2 8x 16 n2 16 n 64
• n2 16n 64

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Factor 9g2 12g 4

• Factor first term 3g , 3g
• Factor last term 2, 2
• Answer (3g 2)(3g 2) (3g 2)2
• Try
• 9g2 12g 4
• 4 t2 36t 81
• 4 t2 - 36t 81

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How do we factor the difference of two squares?

• Factor x2 64.
• What are the factors of x2? And of 64?
• x, x, 8, 8
• Remember the b term must cancel out.
• Answer ( x 8)(x 8)

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Practice

• Factor each expression. Check your answer.
• 1. x2 36
• 2. m2 100
• 3. p2 - 49

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• Factor 4x2 121.
• Express 4 as (2x)2 and 121 as 112
• How can we factor this expression?
• Answer (2x 11) (2x 11)

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Practice

• Factor each expression.
• 9v2 4
• 25x2 64
• 4w2 - 49

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Factoring Out a Common Factor

• 10x2 40
• 10 and 40 are not perfect squares. So, try
  factoring the GCF of both terms.
• 10 (x2 4) Now we can factor.
• Answer 10 ( x 2 ) ( x 2)

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Practice

• Factor each expression.
• 8y2 50
• 3c2 75
• 28k2 7

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Summary

• Describe two types of special cases you learned
  in this lesson. Provide an example for each and
  explain how to factor.