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Alg2 3.4 Notes.notebook October 15, 2012
34 Factoring Polynomials
FACTORING REVIEW NEXT 12 SLIDES
*Courtesy of Paul Sullivan
Factoring Review
Why do we factor quadratic equations?Essential Question:
Factor FormStandard Formax2 + bx + c = 0 (x - p)(x - q) = 0
Examplesx2 + 5x + 6 = 0 (x + 2)(x + 3) = 0
(3x + 4)(x - 5) = 03x2 - 11x - 20 = 0
Which are easier to solve?
Alg2 3.4 Notes.notebook October 15, 2012
Learning Targets: We are reviewing how to factor quadratic equations
We will review the 6 different methods for factoring
Method 1: Taking out the common factorMethod 2: Factoring x2 + bx +c by Product and Sum TableMethod 3: Factor by GroupingMethod 4: Factoring ax2 + bx +c by Splitting the Middle TermMethod 5: Special Products - Difference of SquaresMethod 6: Special Product - Perfect Square Quadratic
*Courtesy of Paul Sullivan
Factoring by Taking out the common Factor
7x2(2x2 - 3)Simplify
Factor2 7 x x x x - 3 7 x x
2 7 x x x x - 3 7 x x7 x x (2 x x - 3)
7x2(2x2 - 3)
(Method 1)
14x4 - 21x2
7 x x (2 x x - 3)
2 7 x x x x - 3 7 x x2 7 x x x x - 3 7 x x
Alg2 3.4 Notes.notebook October 15, 2012
1.) 2x2 + 8
Factor each expression by taking out the common factor
2.) 7x2 - 14 3.) 4x2 + 16x - 4
4.) 5x2 + 10 5.) 25x2 - 20 6.) 4x3 + 20x2 - 24x
Solve each equation by factoring
1.) 9x2 + 18x = 0 2.) 8x2 - 12x = 0 3.) 5x2 = 15x
(Method 2) Factoring x2 + bx + c (Product and Sum Table)
x2 + 12x + 20
x2 + 14x + 48
product
sum product
c b
2.)
sum
1 202 104 5
list offactor pairs
2112 9
winning factor pair
(x + 2)(x + 10)
(press for hidden)
1.)
Put in factored formproduct
c bsum x2 + x -303.)
productc b
sum
Alg2 3.4 Notes.notebook October 15, 2012
Factoring by Grouping
x3 + 4x2 + 6x + 24
x3 + 4x2 + 6x + 24first group second group
Step-By-Step1.) Break into two groups.2.) Take out common factor from first group.3.) Take out the common factor from the second group.4.) If the remainder is the same for both groups, factor it out to the first "bubble".5.) The two remaining terms form the second "bubble".
x2(x + 4) + 6x +24
x2(x + 4) + 6(x + 4)
(x + 4)(x2 + 6)
(Method 3):
(press for hidden)
Factor By Grouping
2.) 3x3 - x2 - 21x + 7
5.) 6.)
1.) x3 + 2x2 + 3x + 6
3.) b3 + 5b2 - 24b - 20 4.) c3 + c2 - 12c - 12
Solve each equation by factoring
Alg2 3.4 Notes.notebook October 15, 2012
Factoring ax2 + bx + c by Splitting the Middle Term *British Method
Ex 1: Factor 2x2 + 11x + 5
a cProduct Sum
b
Step by Step1. Make a Product and Sum Table the Product has to be a c. The sum has to be b.
2.) Split the middle term to get four terms.
3.) Factor by grouping
(Method 4)
More Examples Step by Step1. Make a Product and Sum Table the Product has to be a c. The sum has to be b.
2.) Split the middle term to get four terms.
3.) Factor by grouping
a cProduct Sum
b1.) 3x2 - 4x - 7
2.) 6x2 - 19x + 15
Alg2 3.4 Notes.notebook October 15, 2012
Factoring Special Products
Difference of Squares Pattern (Sum and Difference Pattern)
(a + b)(a - b) a2 - b2
Perfect Square Pattern(a + b)2 = (a + b)(a + b) a2 + 2ab + b2
(a - b)2 = (a - b)(a - b) a2 - 2ab + b2
Special Products Patterns (Short Cuts to FOIL)
Sum and Difference Pattern (Difference of Squares)(a - b)(a + b) = a 2 - b2
Perfect Square Pattern (Square of a Binomial)(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
These are examples of the two kinds of special products
Write each expression in simplest form
1.) (x + 4)(x - 4) 2.) (x + 7)2 3.) (3x -5)2
Factor each difference of squares
1.) m2 - 25 2.) 4p2 - 81 3.) 25 - 49x2
4.) 49v2 - 144 4.) 81v2 + 100
Press here for more practice problems
(Method 5)
Alg2 3.4 Notes.notebook October 15, 2012
Factor each Perfect Square Trinomial
1.) x2 + 16x + 64 2.) x2 - 10x + 25
3.) 4x2 + 36x + 81 4.) 9x2 - 30x + 25
5.) x2 - 14x + 49 6.) x2 + 20x + 100
7.) 49x2 - 56x + 16 8.) 36x2 + 60x + 25
Press here for more problems
(Method 6):
34 Factoring Polynomials
Use the Factor Theorem to determine factors of a polynomial.
Factor the sum and difference of two cubes.
To find real roots of a polynomial.
Alg2 3.4 Notes.notebook October 15, 2012
I. Factor Theorem
1. Determine whether a linear binomial is a factor:A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10)
II. Sum & Difference of Two Cubes
2. Factor 125d3 – 8
(5d – 2)(25d2 + 10d + 4)
Alg2 3.4 Notes.notebook October 15, 2012
III. All together now...3. Factor (grouping):
4. Factor (cubic):
5. Factor (Quadratic):
x3 – x2 – 25x + 25.
4x4 + 108x
4. 4x(x + 3)(x2 – 3x + 9)
3. (x – 1)(x – 5)(x + 5)
5. (x2 + 9)(x2 5)
x4 + 4x2 45
IV. More Fun6. (Homework #28) Factor 24n2 + 3n5
Answers: 6. 3n2(2 + n)(4 2n + n2)7. x4 14x2 32
7. x4 14x2 32
Alg2 3.4 Notes.notebook October 15, 2012
Say I knew (x 7) was a factor of x3 + 3x2 + 2x 504.
How can a write the polynomial as a product?
34 p.177 #2, 3, 5, 7, 11, 15, 17, 19, 25, 27, 31, 36, 38,
39, 41, 42, 51, 53, 54, 56, 57
Check: SA11
