Question

factor the trinomial completely.

\\49p^2 - 28p - 5\\

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

a. \\49p^2 - 28p - 5 = \quad\\ (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials

Step 1: Identify coefficients

For the trinomial \( 49p^2 - 28p - 5 \):

$$ a = 49, \quad b = -28, \quad c = -5 $$

Step 2: Find factors of \( ac \) that add up to \( b \)

Calculate the product \( ac \):

$$ ac = 49 \times (-5) = -245 $$

We need two numbers that multiply to \( -245 \) and add up to \( -28 \):

$$ -35 \times 7 = -245 $$

$$ -35 + 7 = -28 $$

Step 3: Rewrite and factor by grouping

Rewrite the middle term \( -28p \) using \( -35p \) and \( 7p \):

$$ 49p^2 - 35p + 7p - 5 $$

Group the terms:

$$ (49p^2 - 35p) + (7p - 5) $$

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

$$ 7p(7p - 5) + 1(7p - 5) $$

Factor out the common binomial \( (7p - 5) \):

$$ (7p - 5)(7p + 1) $$

Answer:

A. \( 49p^2 - 28p - 5 = (7p - 5)(7p + 1) \)