To factor the given expression [tex]\( x^2 + 16x + 64 \)[/tex], let's first understand the type of expression we are dealing with and then break it down step by step.
1. Identify the quadratic expression:
[tex]$ x^2 + 16x + 64 $[/tex]
2. Check for the simplest factorization:
We begin with the middle term in the quadratic expression [tex]\(16x\)[/tex] and see if we can come up with two numbers whose product equals the constant term [tex]\(64\)[/tex] and sum equals the coefficient of the linear term [tex]\(16\)[/tex].
3. Find factors of [tex]\(64\)[/tex] that add up to [tex]\(16\)[/tex]:
- The constant term is [tex]\(64\)[/tex].
- We need to find two numbers that multiply to [tex]\(64\)[/tex] and add up to [tex]\(16\)[/tex].
4. Possible pairs:
- [tex]\( 1 \times 64 = 64\)[/tex], but [tex]\( 1 + 64 ≠ 16 \)[/tex]
- [tex]\( 2 \times 32 = 64\)[/tex], but [tex]\( 2 + 32 ≠ 16 \)[/tex]
- [tex]\( 4 \times 16 = 64\)[/tex], but [tex]\( 4 + 16 ≠ 16 \)[/tex]
- [tex]\( 8 \times 8 = 64\)[/tex], and [tex]\( 8 + 8 = 16 \)[/tex]
The numbers [tex]\(8\)[/tex] and [tex]\(8\)[/tex] fit perfectly because their product is [tex]\(64\)[/tex] and their sum is [tex]\(16\)[/tex].
5. Rewrite the quadratic expression using these factors:
[tex]$ x^2 + 16x + 64 = (x + 8)(x + 8) $[/tex]
6. Express as a perfect square:
Since both the factors are the same:
[tex]$ (x + 8)(x + 8) = (x + 8)^2 $[/tex]
Thus, the factored form of the given expression [tex]\( x^2 + 16x + 64 \)[/tex] is:
[tex]$ (x + 8)^2 $[/tex]
Correct answer:
C. [tex]\( (x + 8)^2 \)[/tex]
