Question

x² - 14x - 32
a the factors should both be negative.
b one of the factors is (x − 16).

Explanation:

To determine the correct option, we factor the quadratic expression \(x^{2}-14x - 32\).

Step 1: Find two numbers

We need two numbers that multiply to \(-32\) (the constant term) and add up to \(-14\) (the coefficient of the \(x\)-term).
Let's list the factor pairs of \(-32\):

• \(1\times(-32)=-32\) and \(1+(-32)=-31\)
• \(2\times(-16)=-32\) and \(2+(-16)=-14\)

Step 2: Factor the quadratic

Using the numbers \(2\) and \(-16\), we can rewrite the middle term:
\[

$$\begin{align*} x^{2}-14x - 32&=x^{2}+2x-16x - 32\\ &=x(x + 2)-16(x + 2)\\ &=(x + 2)(x-16) \end{align*}$$

\]

Now let's analyze the options:

• Option A: The factors are \((x + 2)\) and \((x-16)\). One factor has a positive constant term and the other has a negative constant term. So the statement "The factors should both be negative" is incorrect.
• Option B: From the factored form \((x + 2)(x-16)\), we can see that one of the factors is \((x-16)\). This statement is correct.

For Option A: Incorrect.
For Option B: Correct.

So the answer is B. One of the factors is \((x - 16)\).

Answer:

To determine the correct option, we factor the quadratic expression \(x^{2}-14x - 32\).

Step 1: Find two numbers

We need two numbers that multiply to \(-32\) (the constant term) and add up to \(-14\) (the coefficient of the \(x\)-term).
Let's list the factor pairs of \(-32\):

• \(1\times(-32)=-32\) and \(1+(-32)=-31\)
• \(2\times(-16)=-32\) and \(2+(-16)=-14\)

Step 2: Factor the quadratic

Using the numbers \(2\) and \(-16\), we can rewrite the middle term:
\[

$$\begin{align*} x^{2}-14x - 32&=x^{2}+2x-16x - 32\\ &=x(x + 2)-16(x + 2)\\ &=(x + 2)(x-16) \end{align*}$$

\]

Now let's analyze the options:

• Option A: The factors are \((x + 2)\) and \((x-16)\). One factor has a positive constant term and the other has a negative constant term. So the statement "The factors should both be negative" is incorrect.
• Option B: From the factored form \((x + 2)(x-16)\), we can see that one of the factors is \((x-16)\). This statement is correct.

For Option A: Incorrect.
For Option B: Correct.

So the answer is B. One of the factors is \((x - 16)\).