To determine which factoring method can be applied to the polynomial [tex]\( x^2 + 12x + 36 \)[/tex], let's first understand the different types of factoring methods provided:
1. Perfect-square trinomial: This involves a trinomial of the form [tex]\( a^2 + 2ab + b^2 \)[/tex], which can be factored into [tex]\( (a + b)^2 \)[/tex].
2. Difference of squares: This involves a binomial of the form [tex]\( a^2 - b^2 \)[/tex], which can be factored into [tex]\( (a + b)(a - b) \)[/tex].
3. Sum of cubes: This involves a binomial of the form [tex]\( a^3 + b^3 \)[/tex], which can be factored into [tex]\( (a + b)(a^2 - ab + b^2) \)[/tex].
4. Difference of cubes: This involves a binomial of the form [tex]\( a^3 - b^3 \)[/tex], which can be factored into [tex]\( (a - b)(a^2 + ab + b^2) \)[/tex].
Given the polynomial [tex]\( x^2 + 12x + 36 \)[/tex]:
1. Perfect-square trinomial:
Let's check if it fits the perfect-square trinomial form:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Here, we identify:
[tex]\[
a = x \quad \text{and} \quad b = 6
\][/tex]
Substituting these values:
[tex]\[
(x + 6)^2 = x^2 + 2(x)(6) + 6^2 = x^2 + 12x + 36
\][/tex]
The polynomial [tex]\( x^2 + 12x + 36 \)[/tex] matches the form [tex]\( a^2 + 2ab + b^2 \)[/tex], thus it can be factored as [tex]\( (x + 6)^2 \)[/tex].
2. Difference of squares:
The form [tex]\( a^2 - b^2 \)[/tex] requires both terms to be squares and separated by a subtraction. Our polynomial has three terms and a plus sign, so it doesn't match this form.
3. Sum of cubes:
The form [tex]\( a^3 + b^3 \)[/tex] requires both terms to be cubes. Our polynomial has three terms involving squares and linear terms, so it doesn't match this form.
4. Difference of cubes:
The form [tex]\( a^3 - b^3 \)[/tex] requires both terms to be cubes and separated by a subtraction. Our polynomial has three terms and a plus sign, so it doesn't match this form.
Given this analysis, the correct factoring method for the polynomial [tex]\( x^2 + 12x + 36 \)[/tex] is the perfect-square trinomial.
