Chapter 6 FaCtoring and the distributive ProPerty 157- x^2 + 4x − 21 =
- x^2 + 13x + 36 =
- x^2 + 5x − 24 =
✔SOLUTIONS
- x^2 − 5x − 6 = (x − 6)(x + 1)
- x^2 + 2x + 1 = (x + 1)(x + 1) = (x + 1)^2
- x^2 + 3x − 10 = (x + 5)(x − 2)
- x^2 − 6x + 8 = (x − 4)(x − 2)
- x^2 − 11x − 12 = (x − 12)(x + 1)
- x^2 − 9x + 14 = (x − 7)(x − 2)
- x^2 + 7x + 10 = (x + 5)(x + 2)
- x^2 + 4x − 21 = (x + 7)(x − 3)
- x^2 + 13x + 36 = (x + 4)(x + 9)
- x^2 + 5x − 24 = (x + 8)(x − 3)
We can use a factoring shortcut when the first term is x^2. If the second sign
is plus, choose the factors whose sum is the coefficient of the second term.
For example, in the expression x^2 – 7x + 6 we need the factors of 6 to sum to
7: x^2 – 7x + 6 = (x – 1)(x – 6). The factors of 6 we need for x^2 + 5x + 6 need to
sum to 5: x^2 + 5x + 6 = (x + 2)(x + 3).
If the second sign is minus, the difference of the factors needs to be the coef-
ficient of the middle term. If the first sign is plus, the bigger factor has the plus
sign. If the first sign is minus, the bigger factor has the minus sign.
EXAMPLES
Factor the quadratic polynomial.x^2 + 3x – 10: The factors of 10 whose difference is 3 are 2 and 5. The first
sign is plus, so the plus sign goes with 5, the bigger factor:
x^2 + 3x – 10 = (x + 5)(x – 2).
x^2 – 5x – 14: The factors of 14 whose difference is 5 are 2 and 7. The first
sign is minus, so the minus sign goes with 7, the bigger
factor: x^2 – 5x – 14 = (x – 7)(x + 2).EXAMPLES
Factor the quadratic polynomial.