Algebra II: Factoring

Factoring ax ² + bx + c

Factoring ax ² + bx + c

This section explains how to factor expressions of the form ax ² + bx + c , where a , b , and c are integers.

First, factor out all constants which evenly divide all three terms. If a is negative, factor out -1. This will leave an expression of the form d (ax ² + bx + c) , where a , b , c , and d are integers, and a > 0 . We can now turn to factoring the inside expression.

Here is how to factor an expression ax ² + bx + c , where a > 0 :

1. Write out all the pairs of numbers that, when multiplied, produce a .
2. Write out all the pairs of numbers that, when multiplied, produce c .
3. Pick one of the a pairs -- (a ₁, a ₂) -- and one of the c pairs -- (c ₁, c ₂) .
4. If c > 0 : Compute a ₁ c ₁ + a ₂ c ₂ . If | a ₁ c ₁ + a ₂ c ₂| = b , then the factored form of the quadratic is
   1. (a ₁ x + c ₂)(a ₂ x + c ₁) if b > 0 .
   2. (a ₁ x - c ₂)(a ₂ x - c ₁) if b < 0 .
5. If a ₁ c ₁ + a ₂ c ₂≠b , compute a ₁ c ₂ + a ₂ c ₁ . If a ₁ c ₂ + a ₂ c ₁ = b , then the factored form of the quadratic is (a ₁ x + c ₁)(a ₂ x + c ₂) or (a ₁ x + c ₁)(a ₂ x + c ₂) . If a ₁ c ₂ + a ₂ c ₁≠b , pick another set of pairs.
6. If c < 0 : Compute a ₁ c ₁ -a ₂ c ₂ . If | a ₁ c ₁ - a ₂ c ₂| = b , then the factored form of the quadratic is:
   
   (a ₁ x - c ₂)(a ₂ x + c ₁) where a ₁ c ₁ > a ₂ c ₂ if b > 0 and a ₁ c ₁ < a ₂ c ₂ if b < 0 .

Using FOIL, the outside pair plus (or minus) the inside pair must equal b .

1. Check.

Example 1: Factor 3x ² - 8x + 4 .

1. Numbers that produce 3: (1, 3).
2. Numbers that produce 4: (1, 4), (2, 2).
3. • (1, 3) and (1, 4): 1(1) + 3(4) = 11≠8 . 1(4) + 3(1) = 7≠ = 8 .
   • (1, 3) and (2, 2): 1(2) + 3(2) = 8 .
   • (x - 2)(3x - 2) .
4. Check: (x - 2)(3x - 2) = 3x ² -2x - 6x + 4 = 3x ² - 8x + 4 .

Example 2: Factor 12x ² + 17x + 6 .

1. Numbers that produce 12: (1, 12), (2, 6), (3, 4).
2. Numbers that produce 6: (1, 6), (2, 3).
3. • (1, 12) and (1, 6): 1(1) + 12(6) = 72 . 1(6) + 12(1) = 18 .
   • (1, 12) and (2, 3): 1(2) + 12(3) = 38 . 1(3) + 12(2) = 27 .
   • (2, 6) and (1, 6): 2(1) + 6(6) = 38 . 2(6) + 6(1) = 18 .
   • (2, 6) and (2, 3): 2(2) + 6(3) = 22 . 2(3) + 6(2) = 18 .
   • (3, 4) and (1, 6): 3(1) + 4(6) = 27 . 3(6) + 4(1) = 22 .
   • (3, 4) and (2, 3): 3(2) + 4(3) = 18 . 3(3) + 4(2) = 17 .
   (3x + 2)(4x + 3) .
4. Check: (3x + 2)(4x + 3) = 12x ² +9x + 8x + 6 = 12x ² + 17x + 6 .

Example 3: Factor 4x ² - 5x - 21 .

1. Numbers that produce 4: (1, 4), (2, 2).
2. Numbers that produce 21: (1, 21), (3, 7).
3. • (1, 4) and (1, 21): 1(1) -4(21) = - 83 . 1(21) - 4(1) = 17 .
   • (1, 4) and (3, 7): 1(3) - 4(7) = - 25 . 1(7) - 4(3) = - 5 .
   (x - 3)(4x + 7) .
4. Check: (x - 3)(4x + 7) = 4x ² +7x - 12x - 21 = 4x ² - 5x - 21 .

Any easy way to check your answer is to pick a value of x ( x = 10 is a good value to pick) and evaluate the original expression and the factored form. You should get the same result for both expressions.