Question

completely factor the trinomial, if possible.

\\8x^2 - 63x - 8\\

Explanation:

Identify the coefficients

We begin with the given quadratic trinomial:

$$8x^2 - 63x - 8$$

Using the Factoring Trinomials knowledge point, we identify the coefficients by comparing the expression to the standard form \(ax^2 + bx + c\):

• \(a = 8\)
• \(b = -63\)
• \(c = -8\)

Find the key product and sum

Using the Finding Factors with Given Sum and Product knowledge point, we need to find two integers whose product is \(a \cdot c\) and whose sum is \(b\):

• Product: \(a \cdot c = 8 \cdot (-8) = -64\)
• Sum: \(b = -63\)

We look for two numbers that multiply to \(-64\) and add up to \(-63\). These numbers are \(-64\) and \(1\), since:

$$(-64) \cdot 1 = -64$$

$$-64 + 1 = -63$$

Rewrite the middle term

Using the Trinomial Factoring by Grouping knowledge point, we split the middle term \(-63x\) using the two numbers we found:

$$8x^2 - 64x + x - 8$$

Factor by grouping

Now, we group the terms into two pairs and factor out the greatest common factor from each pair:

$$(8x^2 - 64x) + (x - 8)$$

Factor out \(8x\) from the first group:

$$8x(x - 8) + 1(x - 8)$$

Both terms now share a common binomial factor of \((x - 8)\). We factor this out:

$$(8x + 1)(x - 8)$$

Answer:

\((8x + 1)(x - 8)\)