Question

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

a. (2m^3 + 54 =) (factor completely. simplify your answer.)
b. the polynomial is prime.

Explanation:

🆕 New Concept Discovered: Factoring Sum of Cubes
Breaking down binomials with perfect cubes

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

First, look for any common factors in both terms of the expression:

$$ 2m^3 + 54 $$

Both terms are divisible by \(2\). Factoring out \(2\) gives:

$$ 2(m^3 + 27) $$

Step 2: Identify the sum of cubes

The expression inside the parentheses, \(m^3 + 27\), is a sum of two perfect cubes:

• \(m^3\) is the cube of \(m\), since \((m)^3 = m^3\)
• \(27\) is the cube of \(3\), since \((3)^3 = 27\)

We can rewrite this as:

$$ 2(m^3 + 3^3) $$

Step 3: Apply the sum of cubes formula

The formula for factoring a sum of cubes is:

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

Using \(a = m\) and \(b = 3\):

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

$$ m^3 + 3^3 = (m + 3)(m^2 - 3m + 9) $$

Step 4: Write the complete factored form

Combine the GCF from Step 1 with the factored binomial:

$$ 2(m + 3)(m^2 - 3m + 9) $$

Answer:

A. \(2m^3 + 54 = 2(m + 3)(m^2 - 3m + 9)\)