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UNIT 7: FACTORING POLYNOMIALS

7-1 Factoring GCF

Greatest Common Factor

The biggest number that can go into all the

given numbers

To factor, find the GCF then do the

distributive property backwards

Find the GCF of

1. 6𝑚3𝑛 𝑎𝑛𝑑22𝑚2𝑛2

Factor each polynomial:

2. 4x2 −8x

3. 25a4 + 15a2

4. 28a2b + 56abc2

5. 20x2 − 24xy2 +12y

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Factoring Special Cases

7-2 Difference of Squares

Difference of Squares

Perfect square minus a perfect square

-Two terms

-Both perfect squares

-Minus between them

2 2a b a b a b

Always look to factor completely!!!

Factor each polynomial completely:

1. x2 − 25

2. a2 − 121

3. 16y4 − z4

4. 32x2 − 50y4

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5. a4 − 81b8

6. 9a2 +4

7. 16x12 − 1

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Factoring Special Cases

7-3 Factoring Trinomial Squares

Perfect Square Trinomials

22 22a ab b a b

22 22a ab b a b

Always look to factor completely

Factor each polynomial completely:

7. x2 + 12x + 36

8. a2 − 14a + 49

9. 49a2 − 42ab + 9b2

10. 9a2 + 24ab + 16b2

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7-4 Factoring Trinomials

To factor:

Use Guess & Check

Use the “product sum” X

ax2 +bx+c

product

a c

b

sum

*Check your answer by

multiplying (FOIL)

Factor each trinomial:

1. x2 + 7x + 10

2. y2 − 13y + 48

3. m2 − m − 90

4. x2 + 3x − 54

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To factor:

1. First look for the GCF (greatest common

factor)

2. Use guess & check or the “product sum”

X method.

5. 3t2 − 3t − 60

6. 2ab2 − 8ab + 8a

7. −p2 + 15p +54

8. 4w3 − 28w2 −120w

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7-5 More Factoring Trinomials

1. Look for the GCF

2. Use guess & check or the “product

sum” X method

product

a c

b

sum

3. Put it in the box

Factor each trinomial completely:

1. 2x2 + 9x + 10

2. 3y2 + 13y + 4

3. 20a2 − 21a − 5

4. 2x2 − 5x + 2

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5. 2c2 + 5c − 2

6. 6x2 − 14x − 12

7. 8x2 − 10x − 12

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7-6 Factoring by Grouping

When factoring 4 term polynomials we use

factor by grouping.

Always look to Factor completely.

Examples

1. x3 + 3x2 + 4x + 12

2. 5x4 + 20x3 + 6x + 12

3. 12x4 + 10x3 − 36x2 − 30x

4. 𝑥3 + 2𝑥2 − 𝑥 − 2

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7-7 SOLVING EQUATIONS BY FACTORING

Zero Product Property

For all numbers a & b

If: 0ab ,

Then one of these is true:

1. a = 0, 2. b = 0, 3. Both a and b = 0

FACTORING and SOLVING

Use the following steps to solve equations

using the principle of zero

1. Get zero on one side of the equation by

using the addition property

2. Factor the expression on the other side

of the equation

3. Set each factor equal to zero

4. Solve each equation.

5. Check your solutions.

1. (5x+1)(x-7) = 0

2. 𝑥2 − 𝑥 − 6 = 0

3. 𝑥2 − 6𝑥 = 16

4. 𝑥3 − 3𝑥2 = 28𝑥

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The answers found to solving the equations

are called the root of the polynomial.

5. 2𝑥2 + 3𝑥 = 5

6. 𝑥(𝑥 − 10) = 56

5. Find the roots of 4𝑥2 − 25

6. Find the roots of 𝑥2 + 6𝑥 + 9

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7-8 SOLVE PROBLEMS BY WRITING & SOLVING EQUATIONS

Examples Translate to an equation and solve.

1. The product of one more than a number and one less than the number is 8. Find the

number.

2. The square of a number minus twice the number is 48. Find the number.

Try This. Translate to an equation and solve.

a. The product of seven less than a number and eight less than the number is 0.

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b. The square of a number minus the number is 20.

c. One more than twice the square of a number is 73.

3. The area of the foresail on a 12-meter racing yacht is 93.75 square meters. The sail’s

height is 8.75 meters greater than its base. Find its base and height.

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4. The product of two consecutive integers is 156. Find the integers.

Try This

d. The width of a rectangular card is 2cm less than the length. The area is 15 cm squared.

Find the length and width?