Question

factor the trinomial completely. do not forget to factor out the gcf first.

\\4x^3 + 20x^2 + 16x\\

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

a. \\4x^3 + 20x^2 + 16x = \\
(type your answer in factored form.)

b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring monomials · factoring trinomials

Step 1: Factor out the GCF

Find the Greatest Common Factor (GCF) of the terms in \( 4x^3 + 20x^2 + 16x \).

The GCF of the coefficients \( 4, 20, \) and \( 16 \) is \( 4 \).
The GCF of the variable parts \( x^3, x^2, \) and \( x \) is \( x \).

Factor out \( 4x \):

$$ 4x^3 + 20x^2 + 16x = 4x(x^2 + 5x + 4) $$

Step 2: Factor the remaining trinomial

Factor the quadratic trinomial \( x^2 + 5x + 4 \).

Find two numbers that multiply to \( 4 \) and add to \( 5 \). These numbers are \( 4 \) and \( 1 \).

$$ x^2 + 5x + 4 = (x + 4)(x + 1) $$

Step 3: Write the completely factored expression

Combine the GCF and the factored trinomial:

$$ 4x(x + 4)(x + 1) $$

Answer:

A. \( 4x^3 + 20x^2 + 16x = 4x(x + 4)(x + 1) \)