Question

factor the trinomial completely.

(a^2 - 7ab + 12b^2)

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

a. (a^2 - 7ab + 12b^2 =) (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials

Step 1: Identify the coefficients

We need to factor the trinomial:

$$ a^2 - 7ab + 12b^2 $$

This is a quadratic-like expression in terms of \( a \) and \( b \). We look for two numbers that:

• Multiply to the coefficient of the last term: \( 12 \)
• Add to the coefficient of the middle term: \( -7 \)

Step 2: Find the factors

The two numbers that multiply to \( 12 \) and add to \( -7 \) are \( -3 \) and \( -4 \):

$$ (-3) \cdot (-4) = 12 $$

$$ (-3) + (-4) = -7 $$

Step 3: Write the factored form

Using these factors, we write the expression as:

$$ (a - 3b)(a - 4b) $$

Answer:

A. \( a^2 - 7ab + 12b^2 = (a - 3b)(a - 4b) \)