To factor the polynomial [tex]\(m^2 + 12m + 36\)[/tex], follow these steps:
1. Identify the polynomial form: The given polynomial is [tex]\(m^2 + 12m + 36\)[/tex]. This is a quadratic polynomial which can be factored into the form [tex]\((m+a)(m+b)\)[/tex] if it factors nicely.
2. Recognize a perfect square: In this instance, observe that the polynomial [tex]\(m^2 + 12m + 36\)[/tex] can be recognized as a perfect square trinomial.
3. Perfect square formula: Remember that a perfect square trinomial follows the form [tex]\(x^2 + 2xy + y^2 = (x + y)^2\)[/tex]. In this case:
- Here, [tex]\(m^2\)[/tex] is the square of [tex]\(m\)[/tex].
- [tex]\(36\)[/tex] is the square of [tex]\(6\)[/tex] (because [tex]\(6^2 = 36\)[/tex]).
- [tex]\(12m\)[/tex] is twice the product of [tex]\(m\)[/tex] and [tex]\(6\)[/tex] (because [tex]\(2 \cdot m \cdot 6 = 12m\)[/tex]).
4. Express the polynomial as a perfect square: Based on the form, we see that [tex]\(m^2 + 12m + 36\)[/tex] fits the formula [tex]\((m + 6)^2\)[/tex].
Therefore, the factored form of the polynomial [tex]\(m^2 + 12m + 36\)[/tex] is:
[tex]\[
(m + 6)^2
\][/tex]
So, the factorization is:
[tex]\[
m^2 + 12m + 36 = (m + 6)^2
\][/tex]
