Question

factor.
$5z^2 -19z + 18$

Explanation:

To factor the quadratic expression \(5z^2 - 19z + 18\), we use the method of factoring by grouping.

Step 1: Multiply the coefficient of \(z^2\) and the constant term

The coefficient of \(z^2\) is \(5\) and the constant term is \(18\). Their product is \(5\times18 = 90\).

Step 2: Find two numbers that multiply to \(90\) and add up to \(-19\) (the coefficient of \(z\))

We need two numbers that multiply to \(90\) and add up to \(-19\). The numbers are \(-9\) and \(-10\) because \((-9)\times(-10)=90\) and \(-9 + (-10)=-19\).

Step 3: Rewrite the middle term using these two numbers

Rewrite \(-19z\) as \(-9z - 10z\):

$$ 5z^2 - 9z - 10z + 18 $$

Step 4: Group the terms and factor by grouping

Group the first two terms and the last two terms:

$$ (5z^2 - 9z) + (-10z + 18) $$

Factor out the greatest common factor (GCF) from each group:
From the first group \(5z^2 - 9z\), the GCF is \(z\), so we get \(z(5z - 9)\).
From the second group \(-10z + 18\), the GCF is \(-2\), so we get \(-2(5z - 9)\).

Now we have:

$$ z(5z - 9) - 2(5z - 9) $$

Step 5: Factor out the common binomial factor

The common binomial factor is \((5z - 9)\), so we factor that out:

$$ (5z - 9)(z - 2) $$

Answer:

\(\boxed{(5z - 9)(z - 2)}\)