1Cademy - Factoring $$r^2 - 8rs

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Factoring Trinomials of the Form $x^2 + bxy + cy^2$

Example

Factoring $r^2 - 8rs - 9s^2$

Factor $r^2 - 8rs - 9s^2$ by applying the two-variable trinomial factoring strategy. Because the first term is $r^2$ , each binomial factor begins with $r$ . Because the last term contains $s^2$ , the second term of each binomial must include $s$ . The last term of the trinomial is negative ( $-9s^2$ ), so the factors must have opposite signs — one positive and one negative.

Step 1 — Set up two binomials: $(r\_s)(r\_s)$ , where each blank will be filled with a coefficient of $s$ , and the signs will be opposite.

Step 2 — Find two numbers that multiply to $-9$ and add to $-8$ . List the factor pairs of $-9$ and check their sums:

Factors of $-9$	Sum of factors
$1, -9$	$1 + (-9) = -8$ ✓
$-1, 9$	$-1 + 9 = 8$
$3, -3$	$3 + (-3) = 0$

The pair $1$ and $-9$ has a product of $-9$ and a sum of $-8$ .

Step 3 — Use $1$ and $-9$ as the coefficients of $s$ in the last terms: $(r + s)(r - 9s)$ .

Step 4 — Check by multiplying: $(r + s)(r - 9s) = r^2 - 9rs + rs - 9s^2 = r^2 - 8rs - 9s^2$ ✓.

The factored form is $(r + s)(r - 9s)$ . This example extends the two-variable trinomial factoring strategy to a case where the coefficient of $s^2$ is negative, requiring the binomial factors to use opposite signs — one addition and one subtraction — just as in the single-variable case when $c$ is negative.

Updated 2026-04-29

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