Question

factor the following sum of two cubes.

(z^3 + 125)

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

a. (z^3 + 125 = \quad) (factor completely. simplify your answer.)
b. the polynomial is prime.

Explanation:

🆕 New Concept Discovered: Factoring Sum and Difference of Cubes
Splitting cubic expressions into linear and quadratic parts

Step 1: Identify the terms as perfect cubes

To factor a sum of two cubes, we first rewrite each term in the expression \( z^3 + 125 \) as a perfect cube:

• The first term is \( z^3 \), which is \( (z)^3 \).
• The second term is \( 125 \), which is \( (5)^3 \) because \( 5 \times 5 \times 5 = 125 \).

So, we have:

$$ a = z $$

$$ b = 5 $$

Step 2: Apply the sum of cubes formula

The algebraic formula for factoring the sum of two cubes is:

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

Substituting \( a = z \) and \( b = 5 \) into this formula gives:

$$ z^3 + 125 = (z + 5)(z^2 - (z)(5) + 5^2) $$

Step 3: Simplify the terms

Simplify the terms inside the second set of parentheses:

• \( (z)(5) = 5z \)
• \( 5^2 = 25 \)

Putting it all together:

$$ z^3 + 125 = (z + 5)(z^2 - 5z + 25) $$

Answer:

A. \( z^3 + 125 = (z + 5)(z^2 - 5z + 25) \)