Question

factor completely.

(42m^2 - 13m - 42)

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

a. (42m^2 - 13m - 42 = \\)

b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials · Factoring by Grouping

Step 1: Identify coefficients and find the product \(ac\)

For the quadratic expression \(42m^2 - 13m - 42\):

$$ a = 42, \quad b = -13, \quad c = -42 $$

$$ ac = 42 \times (-42) = -1764 $$

Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)

We need two numbers that multiply to \(-1764\) and add to \(-13\).

Testing factors of \(1764\) that are close to \(\sqrt{1764} = 42\):

$$ 36 \times (-49) = -1764 $$

$$ 36 + (-49) = -13 $$

The two numbers are \(36\) and \(-49\).

Step 3: Rewrite the middle term and factor by grouping

Rewrite \(-13m\) as \(36m - 49m\):

$$ 42m^2 + 36m - 49m - 42 $$

Group the terms:

$$ (42m^2 + 36m) - (49m + 42) $$

Factor out the greatest common factor (GCF) from each group:

$$ 6m(7m + 6) - 7(7m + 6) $$

Factor out the common binomial \((7m + 6)\):

$$ (6m - 7)(7m + 6) $$

Answer:

A. \(42m^2 - 13m - 42 = (6m - 7)(7m + 6)\)