Question

factor.

\\5z^2 - 23z - 10\\

Explanation:

Identify coefficients of the quadratic expression

We begin with the quadratic expression:

$$5z^2 - 23z - 10$$

The coefficients are:

• \(a = 5\)
• \(b = -23\)
• \(c = -10\)

Find two numbers that multiply to ac and add to b

We calculate the product \(ac\):

$$ac = 5 \times (-10) = -50$$

We need to find two integers that multiply to \(-50\) and add up to \(b = -23\).
Let's test factors of \(-50\):

• \(-25 \times 2 = -50\)
• \(-25 + 2 = -23\)

The two numbers are \(-25\) and \(2\).

Rewrite the middle term and factor by grouping

We split the middle term \(-23z\) using \(-25z\) and \(2z\):

$$5z^2 - 25z + 2z - 10$$

Now, we group the terms into two pairs:

$$(5z^2 - 25z) + (2z - 10)$$

Factor out the greatest common factor (GCF) from each group:

• From \(5z^2 - 25z\), factor out \(5z\): \(5z(z - 5)\)
• From \(2z - 10\), factor out \(2\): \(2(z - 5)\)

This gives:

$$5z(z - 5) + 2(z - 5)$$

Factor out the common binomial

Both terms share a common binomial factor of \((z - 5)\). Factoring it out yields:

$$(5z + 2)(z - 5)$$

Answer:

Factor.
\(5z^2 - 23z - 10\) = <blank>\((5z + 2)(z - 5)\)</blank>