Question

completely factor the trinomial, if possible.

\\2x^2 - 18x + 40\\

Explanation:

Factor out the greatest common factor

We begin by identifying the greatest common factor (GCF) of the terms in the trinomial \(2x^2 - 18x + 40\).

Using the Factoring Polynomials by GCF knowledge point, we find the GCF of the coefficients \(2\), \(-18\), and \(40\), which is \(2\). We factor \(2\) out of the entire expression:

$$ 2x^2 - 18x + 40 = 2(x^2 - 9x + 20) $$

Find factors for the remaining trinomial

Next, we need to factor the quadratic trinomial inside the parentheses, \(x^2 - 9x + 20\).

Using the Factoring Trinomials knowledge point, we look for two integers whose product is the constant term \(20\) and whose sum is the linear coefficient \(-9\).

• The two integers that satisfy these conditions are \(-4\) and \(-5\), since:

$$ (-4) \cdot (-5) = 20 $$

$$ (-4) + (-5) = -9 $$

Write the completely factored expression

We rewrite the trinomial \(x^2 - 9x + 20\) as the product of two binomials using the factors we found:

$$ x^2 - 9x + 20 = (x - 4)(x - 5) $$

Combining this with the GCF factored out in the first step, we obtain the completely factored expression:

$$ 2(x - 4)(x - 5) $$

Answer:

\(2(x - 4)(x - 5)\)