Question

factors and products: constructed response
show all your work in each of the following questions for full value.

1. completely factor each of the following:

a) $5x^2 - 15x - 50$ 1.5 points
b) $2x^2 + 15x + 7$ 2 points
c) $2x^2 - 7xy - 9y^2$ 2 points

Explanation:

Part A: Factoring \( 5x^2 - 15x - 50 \)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \( 5x^2 \), \( -15x \), and \( -50 \) is \( 5 \).
\( 5x^2 - 15x - 50 = 5(x^2 - 3x - 10) \)

Step 2: Factor the quadratic trinomial

We need two numbers that multiply to \( -10 \) and add to \( -3 \). These numbers are \( -5 \) and \( 2 \).
\( x^2 - 3x - 10 = (x - 5)(x + 2) \)

Step 3: Combine the factors

Substitute the factored trinomial back into the expression from Step 1.
\( 5(x^2 - 3x - 10) = 5(x - 5)(x + 2) \)

Step 1: Use the AC method

For \( ax^2 + bx + c \) (here, \( a = 2 \), \( b = 15 \), \( c = 7 \)), find two numbers that multiply to \( a \cdot c = 2 \cdot 7 = 14 \) and add to \( b = 15 \). These numbers are \( 14 \) and \( 1 \).

Step 2: Split the middle term

Rewrite \( 15x \) as \( 14x + x \):
\( 2x^2 + 14x + x + 7 \)

Step 3: Factor by grouping

Group the first two terms and the last two terms:
\( (2x^2 + 14x) + (x + 7) \)
Factor out the GCF from each group:
\( 2x(x + 7) + 1(x + 7) \)

Step 4: Factor out the common binomial

Factor out \( (x + 7) \):
\( (2x + 1)(x + 7) \)

Step 1: Use the AC method (treating \( y \) as a constant)

For \( ax^2 + bx + c \) (here, \( a = 2 \), \( b = -7y \), \( c = -9y^2 \)), find two numbers that multiply to \( a \cdot c = 2 \cdot (-9y^2) = -18y^2 \) and add to \( -7y \). These numbers are \( -9y \) and \( 2y \).

Step 2: Split the middle term

Rewrite \( -7xy \) as \( -9xy + 2xy \):
\( 2x^2 - 9xy + 2xy - 9y^2 \)

Step 3: Factor by grouping

Group the first two terms and the last two terms:
\( (2x^2 - 9xy) + (2xy - 9y^2) \)
Factor out the GCF from each group:
\( x(2x - 9y) + y(2x - 9y) \)

Step 4: Factor out the common binomial

Factor out \( (2x - 9y) \):
\( (x + y)(2x - 9y) \)

Answer:

\( 5(x - 5)(x + 2) \)

Part B: Factoring \( 2x^2 + 15x + 7 \)