1Cademy - Factoring $$8u^2 - 17u - 21$$ Using the ac Method

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Factoring Trinomials of the Form $ax^2 + bx + c$ Using the ac Method

Example

Factoring $8u^2 - 17u - 21$ Using the ac Method

Factor $8u^2 - 17u - 21$ by applying the ac method, demonstrating the process when the constant term is negative and the product $a \cdot c$ is a large negative number.

Step 1 — Factor any GCF. The terms $8u^2$ , $-17u$ , and $-21$ share no common factor, so proceed to the next step.

Step 2 — Find the product $ac$ . Here $a = 8$ and $c = -21$ , so $ac = 8(-21) = -168$ .

Step 3 — Find two numbers that multiply to $-168$ and add to $-17$ . Because the product is negative, the two numbers must have opposite signs, and because the sum is negative, the factor with the larger absolute value must be negative. List factor pairs of $-168$ and check their sums:

Factors of $-168$	Sum of factors
$1, -168$	$1 + (-168) = -167$
$2, -84$	$2 + (-84) = -82$
$3, -56$	$3 + (-56) = -53$
$4, -42$	$4 + (-42) = -38$
$6, -28$	$6 + (-28) = -22$
$7, -24$	$7 + (-24) = -17$ ✓
$8, -21$	$8 + (-21) = -13$

The pair $7$ and $-24$ works: $7 \cdot (-24) = -168$ and $7 + (-24) = -17$ .

Step 4 — Split the middle term $-17u$ into $7u$ and $-24u$ :

$8u^2 - 17u - 21 = 8u^2 + 7u - 24u - 21$

Step 5 — Factor by grouping. Group into two pairs and factor the GCF from each:

$u(8u + 7) - 3(8u + 7)$

Both groups share the common binomial $(8u + 7)$ . Factor it out:

$(8u + 7)(u - 3)$

Step 6 — Check by multiplying: $(8u + 7)(u - 3) = 8u^2 - 24u + 7u - 21 = 8u^2 - 17u - 21$ ✓

The factored form is $(8u + 7)(u - 3)$ . Unlike the earlier example of $6x^2 + 7x + 2$ where $ac$ was positive, here the negative constant produces a negative $ac$ product of $-168$ , which means the two numbers $m$ and $n$ must have opposite signs. Additionally, because the middle coefficient $-17$ is negative, the larger factor (in absolute value) must carry the negative sign. This example also illustrates that the factor-pair search may involve many rows when $|ac|$ is large.

Updated 2026-04-21

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OpenStax Elementary Algebra

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