Question

factor the trinomial completely. if the trinomial is prime, so state.

\\4x^3 - 28x^2 + 48x\\

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

a. \\(4x^3 - 28x^2 + 48x =\\) (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials

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

Identify the GCF of the terms in the expression \( 4x^3 - 28x^2 + 48x \).

The GCF of the coefficients \( 4, -28, \) and \( 48 \) is \( 4 \).
The GCF of the variable parts \( x^3, x^2, \) and \( x \) is \( x \).

Factor out \( 4x \):

$$ 4x(x^2 - 7x + 12) $$

Step 2: Factor the remaining trinomial

Factor the quadratic trinomial \( x^2 - 7x + 12 \) by finding two numbers that multiply to \( 12 \) and add up to \( -7 \).

These numbers are \( -3 \) and \( -4 \):

$$ x^2 - 7x + 12 = (x - 3)(x - 4) $$

Combine this with the GCF from Step 1:

$$ 4x(x - 3)(x - 4) $$

Answer:

A. \( 4x^3 - 28x^2 + 48x = 4x(x - 3)(x - 4) \)