1Cademy - Factoring $$10y^4 + 55y^3 + 60y^2$$

Learn Before

Strategy to Factor Polynomials Completely

Example

Factoring $10y^4 + 55y^3 + 60y^2$

Factor $10y^4 + 55y^3 + 60y^2$ completely by first extracting a monomial GCF that includes a variable, and then factoring the remaining trinomial using the trial and error method for trinomials of the form $ax^2 + bx + c$ .

Step 1 — Check for a GCF. The three terms $10y^4$ , $55y^3$ , and $60y^2$ share a numerical factor of $5$ and a variable factor of $y^2$ (the lowest power of $y$ among all terms). The GCF is $5y^2$ . Factor it out:

$10y^4 + 55y^3 + 60y^2 = 5y^2(2y^2 + 11y + 12)$

Step 2 — Factor the trinomial. The expression $2y^2 + 11y + 12$ is a trinomial with leading coefficient $a = 2 \neq 1$ . The only factor pair of $2y^2$ is $y \cdot 2y$ . The factor pairs of 12 (all positive since every term is positive) are $1, 12$ and $2, 6$ and $3, 4$ . Use the GCF elimination shortcut to discard combinations where a binomial factor contains a common factor. After testing the remaining combinations, the pair $(y + 4)(2y + 3)$ produces the correct middle term: inner product $= 4 \cdot 2y = 8y$ , outer product $= y \cdot 3 = 3y$ , sum $= 11y$ ✓

Step 3 — Write the completely factored form. Remember to include the GCF $5y^2$ :

$5y^2(y + 4)(2y + 3)$

Step 4 — Check by multiplying: $5y^2(y + 4)(2y + 3) = 5y^2(2y^2 + 3y + 8y + 12) = 5y^2(2y^2 + 11y + 12) = 10y^4 + 55y^3 + 60y^2$ ✓

The completely factored form is $5y^2(y + 4)(2y + 3)$ . This example combines two techniques: extracting a monomial GCF with a variable component ( $5y^2$ ), then applying the trial and error method for $ax^2 + bx + c$ to the remaining trinomial. Unlike earlier examples where extracting the GCF reduced the leading coefficient to 1, here the trinomial inside the parentheses still has $a = 2 \neq 1$ , requiring the more involved trial-and-error approach. Always remember to include the GCF in the final factored form.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After