1Cademy - Factoring $$2n^2 - 8n

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Strategy to Factor Polynomials Completely

Example

Factoring $2n^2 - 8n - 42$

Factor $2n^2 - 8n - 42$ completely by first extracting the GCF and then factoring the resulting trinomial.

Step 1 — Check for a GCF: The three terms $2n^2$ , $-8n$ , and $-42$ all share a factor of $2$ . Factor it out: $2(n^2 - 4n - 21)$ .

Step 2 — Classify the expression inside the parentheses: The expression $n^2 - 4n - 21$ is a trinomial with a leading coefficient of 1, so the "undo FOIL" method applies. Set up two binomials: $(n\quad)(n\quad)$ .

Step 3 — Find two numbers whose product is $-21$ and whose sum is $-4$ . List the factor pairs of $-21$ and check their sums:

Factors of $-21$	Sum of factors
$1, -21$	$1 + (-21) = -20$
$3, -7$	$3 + (-7) = -4$ ✓

The pair $3$ and $-7$ works.

Step 4 — Write the fully factored form: $2(n + 3)(n - 7)$ .

Step 5 — Check by multiplying: $2(n + 3)(n - 7) = 2(n^2 - 7n + 3n - 21) = 2(n^2 - 4n - 21) = 2n^2 - 8n - 42$ ✓.

The completely factored form is $2(n + 3)(n - 7)$ . This example illustrates a two-step factoring process: extracting the GCF first reduces the leading coefficient to 1, which then allows the simpler "undo FOIL" trinomial factoring method to be applied to the expression inside the parentheses. Always remember to check for a GCF before attempting other factoring techniques.

Updated 2026-04-21

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OpenStax Elementary Algebra

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