College

Identify and factor expressions.

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

Answer :

To solve the problem of identifying and factoring the polynomial [tex]\(x^3 + 64\)[/tex], let's break it down into steps.

### Step 1: Identify the Type of Polynomial

The expression [tex]\(x^3 + 64\)[/tex] is a type of polynomial known as a "sum of cubes." This is because it can be represented as [tex]\(a^3 + b^3\)[/tex], where:

- [tex]\(a = x\)[/tex]
- [tex]\(b = 4\)[/tex] (since [tex]\(64\)[/tex] is equal to [tex]\(4^3\)[/tex])

### Step 2: Use the Sum of Cubes Factoring Formula

For a sum of cubes, [tex]\(a^3 + b^3\)[/tex], we can use the following factoring formula:

[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]

### Step 3: Apply the Formula

Given that [tex]\(a = x\)[/tex] and [tex]\(b = 4\)[/tex], apply the sum of cubes formula:

1. First Part: [tex]\(a + b = x + 4\)[/tex]

2. Second Part:
- [tex]\(a^2 = x^2\)[/tex]
- [tex]\(-ab = -x \times 4 = -4x\)[/tex]
- [tex]\(b^2 = 4^2 = 16\)[/tex]

Putting it all together from the formula [tex]\((a + b)(a^2 - ab + b^2)\)[/tex], we factor the expression:

[tex]\[
(x + 4)(x^2 - 4x + 16)
\][/tex]

### Conclusion

Thus, the polynomial [tex]\(x^3 + 64\)[/tex] factors to [tex]\((x + 4)(x^2 - 4x + 16)\)[/tex]. This is the factored form of the given expression.