1. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
R.4 Factoring
Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type x2 + bx + c.
Factor trinomials of the type ax2 + bx + c,
a  1, using the FOIL method.
Factor special products of polynomials.
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2. Terms with Common Factors
When factoring, we should always look first to
factor out a factor that is common to all the terms.
Example: x  6x2
= 6 • • 2x  6 • x2
= 6(3 + 2x  x2)
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3. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Factoring by Grouping
In some polynomials, pairs of terms have a common binomial factor that can be removed in the process called factoring by grouping.
Example: x3 + 5x2  10x  50
= (x3 + 5x2) + (10x  50)
= x2(x + 5)  10(x + 5)
= (x2  10)(x + 5)
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4. Trinomials of the Type x2 + bx + c
Factor: x2 + 9x + 14.
Solution:
1. Look for a common factor.
2. Find the factors of 14, whose sum is 9.
Pairs of Factors Sum 1,
2, The numbers we need.
3. The factorization is (x + 2)(x + 7).
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Another Example
Factor: 2y2  20y + 48.
1. First, we look for a common factor.
2(y2  10y + 24)
2. Look for two numbers whose product is 24 and whose sum is 10.
Pairs Sum Pairs Sum
1, 24 25 2, 12 14
3, 8 11 4, 6 10
3. Complete the factorization: 2(y  4)(y  6).
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6. Trinomials of the Type ax2 + bx + c, a  1
FOIL method
1. Factor out the largest common factor.
2. Find two First terms whose product is ax2.
3. Find two Last terms whose product is c.
4. Repeat steps (2) and (3) until a combination is found
for which the sum of the Outside and Inside products
is bx.
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7. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Factor: 8x2 + 10x + 3.
(8x + )(x + )
(8x + 1)(x + 3) middle terms are wrong 24x + x = 25x
(8x + 3)(x + 1) middle terms are wrong 8x + 3x = 11x
(4x + )(2x + )
(4x + 1)(2x + 3) middle terms are wrong 12x + 2x = 14x
(4x + 3)(2x + 1) Correct! 4x + 6x = 10x
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8. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Factor: 12a3  4a2  16a.
1. Factor out the largest common factor, 4a.
4a(3a2  a  4)
2. (3a + )(a + )
(3a + 1)(a  4) middle terms are wrong -12a a = -11a
(3a  1)(a + 4) middle terms are wrong 12a + ( a) = 11a
(3a + 4)(a  1) middle terms are wrong a a = a
(3a  4)(a + 1) correct a + (4a) = -a
4a(3a  4)(a + 1)
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9. Special Factorizations
Difference of Squares
A2  B2 = (A + B)(A  B)
Example
x2  25
= x2 – 52
= (x + 5)(x  5)
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10. Special Factorizations
Squares of Binomials
A2 + 2AB + B2 = (A + B)2
A2  2AB + B2 = (A  B)2
Example
x2 + 12x + 36
= x2 + 2(6) + 62
= (x + 6)2
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11. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
More Factorizations
Sum or Difference of Cubes
A3 + B3 = (A + B)(A2  AB + B2)
A3  B3 = (A  B)(A2 + AB + B2)
Example 8y
= (2y)3 + (5)3
= (2y + 5)(4y2  10y + 25)
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