Question

factor the following sum of two cubes.

(10x^3 + 80)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. (10x^3 + 80 = \\) (factor completely. simplify your answer.)
b. the polynomial is prime.

Explanation:

🆕 New Concept Discovered: Sum of Two Cubes
Factoring using the formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

Step 1: Factor out the greatest common factor (GCF)

First, look for a common factor in both terms of the expression \( 10x^3 + 80 \). Both terms are divisible by \( 10 \):

$$ 10x^3 + 80 = 10(x^3 + 8) $$

Step 2: Identify the sum of cubes

The expression inside the parentheses, \( x^3 + 8 \), is a sum of two perfect cubes because \( 8 = 2^3 \):

$$ x^3 + 8 = x^3 + 2^3 $$

Here, \( a = x \) and \( b = 2 \).

Step 3: Apply the sum of cubes formula

Use the sum of cubes factoring pattern:

$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$

Substitute \( a = x \) and \( b = 2 \) into the formula:

$$ x^3 + 2^3 = (x + 2)(x^2 - 2x + 2^2) $$

$$ x^3 + 2^3 = (x + 2)(x^2 - 2x + 4) $$

Step 4: Write the complete factored expression

Combine the GCF from Step 1 with the factored binomial and trinomial:

$$ 10(x + 2)(x^2 - 2x + 4) $$

Answer:

A. \( 10x^3 + 80 = 10(x + 2)(x^2 - 2x + 4) \)