Question

completely factor the trinomial, if possible.

\\10x^2 - 23x - 5\\

Explanation:

Identify the coefficients of the trinomial

We begin with the given quadratic trinomial:

$$ 10x^2 - 23x - 5 $$

Using the Factoring Trinomials method, we identify the coefficients:

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

Find two numbers with a given sum and product

Using the Finding Factors with Given Sum and Product concept, we need two integers that:

• Multiply to \(a \cdot c = 10 \cdot (-5) = -50\)
• Add up to \(b = -23\)

Let's test factor pairs of \(-50\):

• \(-25\) and \(2\)
• \(-25 \cdot 2 = -50\)
• \(-25 + 2 = -23\)

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

Rewrite the middle term

Using the Trinomial Factoring by Grouping method, we split the middle term \(-23x\) using \(-25x\) and \(2x\):

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

Factor by grouping

Group the terms into two pairs:

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

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

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

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

• From \(2x - 5\), factor out \(1\):

$$ 1(2x - 5) $$

Combine the factored groups:

$$ (5x + 1)(2x - 5) $$

Answer:

\((5x + 1)(2x - 5)\)