2. CHAPTER R: Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
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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 and the grouping method.
Factor special products of polynomials.
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4. 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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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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6. 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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8. 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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Example
Factor: 8x2 + 10x + 3.
(8x + )(x + )
(8x + 1)(x + 3) middle terms are wrong 24x + x = 25x
(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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10. Grouping Method (or ac-method) ax2 + bx + c
1. Factor out the largest common factor.
2. Multiply the leading coefficient a and the constant c.
3. Try to factor the product ac so that the sum of the factors is b. That is, find integers p and q such that pq = ac and p + q = b.
4. Split the middle term. That is, write it as a sum using the factors found in step (3).
5. Factor by grouping.
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Example
Factor: 12a3  4a2  16a.
1. Factor out the largest common factor, 4a.
4a(3a2  a  4)
2. Multiply a and c: (3)(4) = 12.
3. Try to factor 12 so that the sum of the factors is the coefficient of the middle term, 1.
(3)(4) = 12 and 3 + (4) = 1
4. Split the middle term using the numbers found in (3).
3a2 + 3a  4a  4
5. Factor by grouping. 3a2 + 3a  4a  4 = (3a2 + 3a) + (4a  4)
= 3a(a + 1)  4(a + 1)
= (3a  4)(a + 1)
Be sure to include the common factor to get the complete factorization.
4a(3a  4)(a + 1)
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12. Special Factorizations
Difference of Squares
A2  B2 = (A + B)(A  B)
Example
x2  25
= x2 – 52
= (x + 5)(x  5)
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13. 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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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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