Question

question 7 of 15, step 1 of 1

completely factor the given polynomial, if possible. if the polynomial cannot be factored, write nf for your answer.

\\v^2 + v - 56\\

Explanation:

Identify the trinomial coefficients

We begin with the given quadratic expression:

$$v^2 + v - 56$$

This is a trinomial of the form \(av^2 + bv + c\).
Here, the coefficients are:

• \(a = 1\)
• \(b = 1\)
• \(c = -56\)

Find factors of the constant term

Using the Factoring Trinomials knowledge point, we need to find two integers whose product is \(c = -56\) and whose sum is \(b = 1\).
Let these two integers be \(p\) and \(q\).

$$p \cdot q = -56$$

$$p + q = 1$$

Determine the correct integer pair

Since the product is negative, one integer must be positive and the other negative. Since their sum is positive, the integer with the larger absolute value must be positive.
Let's list the factor pairs of \(-56\):

• \(-1 \cdot 56 = -56\) (sum is \(55\))
• \(-2 \cdot 28 = -56\) (sum is \(26\))
• \(-4 \cdot 14 = -56\) (sum is \(10\))
• \(-7 \cdot 8 = -56\) (sum is \(1\))

The correct pair is \(-7\) and \(8\).

Write the factored expression

Since \(a = 1\), we can write the trinomial directly in its factored form using our two integers:

$$(v + p)(v + q)$$

Substituting \(p = -7\) and \(q = 8\):

$$(v - 7)(v + 8)$$

Answer:

\((v - 7)(v + 8)\)