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If a polynomial has three terms, [tex][tex]$x^2 + 12x + 36$[/tex][/tex], which factoring method can be considered?

A. Perfect-square trinomial
B. Difference of squares
C. Sum of cubes
D. Difference of cubes


Sagot :

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.