1Cademy - Factoring $$4y^2

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Strategy to Factor Polynomials Completely

Example

Factoring $4y^2 - 36y + 56$

Factor $4y^2 - 36y + 56$ completely by first extracting the GCF and then factoring the resulting trinomial.

Step 1 — Check for a GCF: The three terms $4y^2$ , $-36y$ , and $56$ all share a factor of $4$ . Factor it out: $4(y^2 - 9y + 14)$ .

Step 2 — Classify the expression inside the parentheses: The expression $y^2 - 9y + 14$ is a trinomial with a leading coefficient of 1, so the "undo FOIL" method applies. The constant term $14$ is positive and the middle coefficient $-9$ is negative, so both numbers in the factor pair must be negative. Set up two binomials: $(y\quad)(y\quad)$ .

Step 3 — Find two negative numbers whose product is 14 and whose sum is $-9$ . List the negative factor pairs of 14 and check their sums:

Factors of $14$	Sum of factors
$-1, -14$	$-1 + (-14) = -15$
$-2, -7$	$-2 + (-7) = -9$ ✓

The pair $-2$ and $-7$ works.

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

Step 5 — Check by multiplying: $4(y - 2)(y - 7) = 4(y^2 - 7y - 2y + 14) = 4(y^2 - 9y + 14) = 4y^2 - 36y + 56$ ✓.

The completely factored form is $4(y - 2)(y - 7)$ . Like the example of factoring $2n^2 - 8n - 42$ , this problem uses a two-step process — extract the GCF first, then factor the resulting trinomial. The difference here is that after extracting the GCF, the remaining trinomial has a positive constant and a negative middle coefficient, which means both numbers in the factor pair must be negative.

Updated 2026-04-21

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OpenStax Elementary Algebra

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