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Unit 11 Factoring and Solving Quadratic

Equations

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11. 1Multiplying a

Monomial by a Polynomial

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NOTES EXAMPLES

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MULTIPLYING A MONOMIAL BY A POLYNOMIAL

monomial- no + or - signsex: 5x2y3

binomial- ONLY 1 + or - signex: 5x3+3y

trinomial- ONLY 2 + or - signsex: 7x2+6x+4

polynomial- 3 or more + or - signsex: 5x2+y2+3x+2y-7

STEPS:1. Multiply the # outside ( ) by each # of each term inside ( )2. Add the exponents of the variables in every term outside ( ) to the same variables in every term inside ( ) 3. Write your answer in standard form (put terms in order from highest to lowest exponent)

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x2y(4x+2xy-8y)12

C

B

A

A

2a(a2-3a+1)

-3x3(2x4-x2-8)

x3(8x2+12x-4)14

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11. 2Multiplication of

Binomials

(Baby Bear - C levels)

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NOTES EXAMPLES

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MULTIPLICATION OF BINOMIALS

BOX MODEL: (2x-4)(x+3)

F.O.I.L: First Outer Inner Last (2x-4)(x+3)

(2x-4)(x+3)

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Examples:

C

C

C

(2a-5)(a+4)

(x-8)(3x-4)

(2m2-6m)(m+2)

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11. 3F.O.I.L.

(Mama Bear - B Levels)

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SQUARINGPLUS/MINUS

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FOIL (Special Cases)

SQUARING BINOMIALS:

- When given a binomial squared, you must re-write to FOIL

ex: (3x-2)2

ex: (4x+3)2

B

B

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Plus/Minus:

- FOIL just like always :-) BUT - notice a pattern that the middle terms will cancel out

ex: (x-2)(x+2)

ex: (3d2+j)(3d2-j)

B

B

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11. 4Multiplying Polynomials

(Papa Bear - A Levels)

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NOTES &

EX #1EX # 2 & EX # 3

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Multiplying Polynomials

Each term from the first binomial should multiply to each term of the trinomial.

Ex1: (2x2-1)(4x2-3x+2) ABox Method:

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Ex2: (4x2+5)(2x2-5x+1)

Ex3: 3x4+(2x2-6)2

A

In HW

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11. 5Factoring the Greatest

Common Factor

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Finding the GCF Factoring a Polynomial

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FINDING THE GREATEST COMMON FACTOR

Finding the GCF:1. 12, 6

2. 9x2, 15x

3. 30x4, 6x6, 18x2

4. x5y3, x8y2

5. 10a3b4c, 5ab3, 20a2b

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Factoring a Polynomial:1. 7x-14

2. 16x-8

3. 5x3+10x2-15x

4. 22x3+x5+14x7

5. 28x2y+14xy2-7xy

C

C

B

B

A

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11. 6Solving by Factoring Out

the Greatest Common Factor

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NOTES

& EX 1Examples

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Solving By Factoring Out the Greatest Common Factor

STEPS:1. Set the equation equal to 0. (A Level only)2. Find the GCF and factor the problem.3. Set each factor equal to 0 and solve.

Ex1: 3x-6=0 C

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B

C

A

Ex2: 2x+10=0

Ex3: 3x2-6x=0

Ex4: -4x2=12x

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11. 8Factor and Solve Special

Trinomials

11. 7Factor and Solving

Trinomials (Quadratics)

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NOTES

& EX 1Examples

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Watch and Learn!x2 + 6x + 8

(__+__)(__+__)

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Factoring & Solving Trinomials (Quadratics)

STEPS:1. Factor out the GCF if there is one (A Level).2. Factor the trinomial using factors of "c" that have a

sum of "b".3. Write as 2 binomials & FOIL to check answer.4. Set each binomial equal to 0 and solve.

Ex 1: x2+9x+18 = 0

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B,C

B,C

A

Ex 2: x2+25x+24 = 0

Ex 3: x2-11x+30 = 0

Ex 4: 5x2-30x+45 = 0

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11. 8Factor and Solve Special

Trinomials

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B Levels A Levels

FOLDED PAPER INSERT (C Levels)

STAPLE

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Factoring & Solving Special Trinomials*DIFFERENCE of squares*

FACTOR & SOLVE: (B Level)

1. 25x2-4=0 2. 16x2-81=0

3. 4x2-16=0 4. 9x2-81=0

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A Level

The area of a rectangle is represented by x2-169

a) Write the expression for the length & width of the rectangle.

L = W =

b) If x=16 ft, what are L & W? L=W=

L

W

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11.9 Factor and Solve by Grouping

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Ex 1 Ex 2 & Ex 3

FOLDED PAPER INSERT (Steps)

STAPLE

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To Factor:1.Look at the terms of the polynomial and see if you can factor out a GCF.2.Group the first two and last two terms of the polynomial together.3.Factor out the GCF separately for the terms grouped together. (At this point you should have 2 matching binomials.)4.The remaining binomials should be the same. If not, see if you can factor out a negative of one of the groups to create common binomials.5.The two GCFs come together to form a binomial and the common binomial is the other factor of the polynomial.

To Solve:1. Set each binomial equal to zero.2. Solve each to find the solutions

Steps: (For polynomials with 4 terms, do not combine the like terms)

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Factor & Solve by Grouping

1. 4x2+12x-x-3=0

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2. 2x2+8x+5x+20=0

3. 9x2+36x-6x-24=0

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11.10Quadratics with Leading

Coefficients >1

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Notes & ExampleExamples

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Quadratics with Leading Coefficients >1

Steps:1. Multiply "a" times "c"2. Find the factors of that product that add to "b"3. Re-write trinomial as polynomial & follow steps from 11.9

1. 4x2+11x-3=0

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2. 3x2+10x-8=0

3. 4x2+26x+40=0

B,C

A

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4. 18x2-30x-3=9x+12

BACK COVER SIDE

A