Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
adding and subtracting rational expressions
identifying common factors
factor the polynomial:
this is a multi - part item.
$x^2 + 5x - 14 = \square\times\square$
$x^2 - 10x + 16 = \square\times\square$

Explanation:

For \( x^2 + 5x - 14 \)

Step1: Find two numbers

We need two numbers that multiply to \( -14 \) (the constant term) and add up to \( 5 \) (the coefficient of \( x \)). Let's list the factor pairs of \( -14 \): \( (7, -2) \) because \( 7\times(-2)=-14 \) and \( 7 + (-2)=5 \).

Step2: Factor the polynomial

Using these two numbers, we can factor \( x^2 + 5x - 14 \) as \( (x + 7)(x - 2) \).

For \( x^2 - 10x + 16 \)

Step1: Find two numbers

We need two numbers that multiply to \( 16 \) (the constant term) and add up to \( -10 \) (the coefficient of \( x \)). The factor pairs of \( 16 \) that add up to \( -10 \) are \( (-8, -2) \) because \( (-8)\times(-2)=16 \) and \( -8 + (-2)=-10 \).

Step2: Factor the polynomial

Using these two numbers, we can factor \( x^2 - 10x + 16 \) as \( (x - 8)(x - 2) \).

Answer:

For \( x^2 + 5x - 14 \): \( (x + 7)(x - 2) \) (so the first box is \( x + 7 \), the second box is \( x - 2 \))
For \( x^2 - 10x + 16 \): \( (x - 8)(x - 2) \) (so the first box is \( x - 8 \), the second box is \( x - 2 \))