Question

factor the trinomial completely.

(25y^2 - 52y + 27)

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

a. (25y^2 - 52y + 27 = \\) (factor completely.)
b. the polynomial is prime.

Explanation:

⚡ Using what you learned: factoring trinomials

Step 1: Identify key values

For the trinomial \( 25y^2 - 52y + 27 \):

$$ a = 25, \quad b = -52, \quad c = 27 $$

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

$$ ac = 25 \times 27 = 675 $$

$$ b = -52 $$

The two numbers are \( -25 \) and \( -27 \):

$$ -25 \times (-27) = 675 $$

$$ -25 + (-27) = -52 $$

Step 2: Rewrite and factor by grouping

Rewrite the middle term using \( -25y \) and \( -27y \):

$$ 25y^2 - 25y - 27y + 27 $$

Group the terms:

$$ (25y^2 - 25y) - (27y - 27) $$

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

$$ 25y(y - 1) - 27(y - 1) $$

Factor out the common binomial \( (y - 1) \):

$$ (25y - 27)(y - 1) $$

Answer:

A. \( 25y^2 - 52y + 27 = (25y - 27)(y - 1) \)