College

The polynomial [tex]$x^3+64$[/tex] is an example of a sum of cubes.

What is the factored form of [tex]$x^3+64$[/tex]?

A. [tex]$\left(x^2+8\right)(x+8)$[/tex]

B. [tex][tex]$(x+4)\left(x^2-4x+16\right)$[/tex][/tex]

C. [tex]$(x-4)\left(x^2+4x+16\right)$[/tex]

Answer :

Let's find the factored form of the polynomial [tex]\( x^3 + 64 \)[/tex].

### Step-by-Step Solution

1. Identify the form of the polynomial:
The polynomial given is [tex]\( x^3 + 64 \)[/tex], which is a sum of cubes. A sum of cubes has the general form [tex]\( a^3 + b^3 \)[/tex].

2. Rewrite the polynomial:
Recognize that [tex]\( 64 \)[/tex] is [tex]\( 4^3 \)[/tex]. So, we can rewrite the polynomial as:
[tex]\[
x^3 + 4^3
\][/tex]

3. Use the sum of cubes formula:
The sum of cubes formula is:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Here, [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex].

4. Apply the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
Substitute [tex]\( a = x \)[/tex] and [tex]\( b = 4 \)[/tex] into the formula:
[tex]\[
(x + 4)((x)^2 - (x)(4) + (4)^2)
\][/tex]

5. Simplify the expression inside the parentheses:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]

So, the factored form of [tex]\( x^3 + 64 \)[/tex] is:
[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]

### Answer
The factored form of [tex]\( x^3 + 64 \)[/tex] is [tex]\((x + 4)(x^2 - 4x + 16)\)[/tex].