Example 7.28 from Elementary Algebra, 7.2 Factor Quadratic Trinomials with Leading Coefficient 1

Factor Quadratic Trinomials with Leading Coefficient 1: example 7.28

Factor: $u^{2} - 9 u v - 12 v^{2}$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We need u in the first term of each binomial and $v$ in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

$1, −12$

Find the numbers that multiply to $−12$ and add to $−9$ .

Factors of $−12$ Sum of factors

$1, −12$ $1 + (−12) = −11$

$−1, 12$ $−1 + 12 = 11$

$2, −6$ $2 + (−6) = −4$

$−2, 6$ $−2 + 6 = 4$

$3, −4$ $3 + (−4) = −1$

$−3, 4$ $−3 + 4 = 1$

Note there are no factor pairs that give us $−9$ as a sum. The trinomial is prime.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Factor: $x^{2} - 7 x y - 10 y^{2}$ .

How did it go?