Question

factor the given trinomial. if the trinomial cannot be factored, indicate
ot factorable\.

\\x^2 + 5x - 14\\

Explanation:

Identify the trinomial coefficients

We begin with the given quadratic expression:

$$x^2 + 5x - 14$$

This is a trinomial of the form \(ax^2 + bx + c\), where:

• \(a = 1\)
• \(b = 5\)
• \(c = -14\)

Find factors of the constant term

We need to find two integers whose product is \(a \cdot c = 1 \cdot (-14) = -14\) and whose sum is \(b = 5\).
Let's list the factor pairs of \(-14\):

• \(-1\) and \(14\) (sum is \(13\))
• \(1\) and \(-14\) (sum is \(-13\))
• \(-2\) and \(7\) (sum is \(5\))
• \(2\) and \(-7\) (sum is \(-5\))

Select the correct factor pair

The pair that multiplies to \(-14\) and adds up to \(5\) is \(-2\) and \(7\):

$$-2 \cdot 7 = -14$$

$$-2 + 7 = 5$$

Write the factored form

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

$$(x - 2)(x + 7)$$

Verify by expanding

We can check our work by multiplying the binomials:

$$(x - 2)(x + 7) = x(x) + 7x - 2x - 2(7) = x^2 + 5x - 14$$

The factored form is correct.

Answer:

\((x - 2)(x + 7)\)