Factoring Trinomials (General)

Learn to factor trinomials when the leading coefficient is not 1 using the AC method.

Advanced25 minLesson

Definition

When factoring a trinomial

ax^2 + bx + c

where

a \neq 1

, we use the AC method (also called the grouping method).

The AC Method

Step 1: Multiply

a \times c

to get the "AC product"

Step 2: Find two numbers that:

Multiply to give AC
Add to give $b$

Step 3: Rewrite the middle term using these two numbers

Step 4: Factor by grouping

ax^2 + bx + c = (px + q)(rx + s)

where

p \times r = a

and

q \times s = c

Try it now

What is the AC product for

2x^2 + 5x + 3

Worked Examples

Factor $2x^2 + 7x + 3$

Identify a, b, and c

$a = 2$ , $b = 7$ , $c = 3$ → Coefficients identified

Calculate AC product

$AC = 2 \times 3 = 6$ → $AC = 6$

Find two numbers that multiply to 6 and add to 7

Factors of 6: $(1, 6)$ , $(2, 3)$ $1 + 6 = 7$ ✓ → Numbers: $1$ and $6$

Rewrite middle term

$2x^2 + 1x + 6x + 3$ → Split $7x$ into $1x + 6x$

Group and factor

$(2x^2 + 1x) + (6x + 3)$ $= x(2x + 1) + 3(2x + 1)$ → Common factor: $(2x + 1)$

Factor out common binomial

$(2x + 1)(x + 3)$ → Factored form

Answer: $(2x + 1)(x + 3)$

Common Mistakes

Forgetting to multiply $a \times c$ and just using $c$

Why it's wrong: When $a \neq 1$ , the product AC is different from just $c$ . For $2x^2 + 7x + 3$ , AC = 6, not 3.

Correct: Always calculate $AC = a \times c$ first. This is the foundation of the AC method.

Incorrect signs when finding factor pairs

Why it's wrong: The signs of the two numbers depend on both AC and $b$ . If AC is negative, the numbers have opposite signs.

Correct: If AC > 0 and b > 0: both positive. If AC > 0 and b < 0: both negative. If AC < 0: opposite signs (larger magnitude matches sign of b).

Grouping terms incorrectly

Why it's wrong: After splitting the middle term, grouping must create a common binomial factor.

Correct: Always verify that both groups yield the same binomial factor before proceeding.

Not checking the answer by expanding

Why it's wrong: It's easy to make sign errors. Always verify by multiplying the factors back out.

Correct: Use FOIL to expand your answer and confirm it equals the original trinomial.

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Practice Problems

16 problems

Problem 1 of 16

Easy

What is the AC product for $2x^2 + 5x + 3$ ?

Why It Matters

Factoring trinomials with leading coefficients other than 1 is essential for:

Solving Quadratic Equations: Many real-world problems lead to equations like $2x^2 + 7x + 3 = 0$
Physics: Projectile motion often involves trinomials with various leading coefficients
Engineering: Optimization problems frequently require factoring complex expressions
Economics: Profit and cost functions are often quadratic with non-unit leading coefficients

The AC method provides a systematic approach that works for any factorable trinomial.

Real World Applications

Projectile Motion

When analyzing the trajectory of a ball thrown upward, the height equation often has a leading coefficient based on gravity.

Example:

The height of a ball is $h = -16t^2 + 48t + 64$ . Factor $-16(t^2 - 3t - 4) = -16(t-4)(t+1)$ to find when the ball hits the ground ( $t = 4$ seconds).

1Try It Yourself

A rocket's height is modeled by $h = -5t^2 + 15t + 20$ meters.

Factor the expression to find when the rocket lands.

Step 1: Write the mathematical expression

First factor out -5: $-5(t^2 - 3t - 4)$

Business Profit Analysis

Companies use quadratic functions to model profit based on production quantity.

Example:

A company's profit is $P = -2x^2 + 14x - 24$ thousand dollars, where $x$ is units in hundreds. Factoring as $-2(x-3)(x-4)$ shows break-even at 300 and 400 units.

2Try It Yourself

A factory's profit model is $P = 3x^2 - 21x + 30$ thousand euros.

Factor to find the break-even production levels.

Step 1: Write the mathematical expression

Factor out the GCF first, then use AC method

Key Takeaways

1The AC method factors trinomials $ax^2 + bx + c$ when $a \neq 1$
2Calculate $AC = a \times c$ , then find two numbers that multiply to AC and add to $b$
3Rewrite the middle term using these numbers, then factor by grouping
4Always verify your answer by expanding (FOIL) the factors
5Look for a GCF first - it simplifies the remaining trinomial

Frequently Asked Questions

If no integer pair multiplies to AC and adds to

b

, the trinomial may be prime (unfactorable over integers) or require the quadratic formula to find roots.

Glossary

AC Method: A factoring technique where you multiply $a \times c$ , find a factor pair, and use grouping
Leading Coefficient: The coefficient $a$ in $ax^2 + bx + c$ , the number in front of $x^2$
Factor by Grouping: Splitting a polynomial into groups and factoring out common factors from each group
Perfect Square Trinomial: A trinomial that factors as $(px + q)^2$ , like $4x^2 - 12x + 9 = (2x-3)^2$

Formula Card

AC Method Steps

ax^2 + bx + c

1. Find $AC = a \times c$ 2. Find numbers $m, n$ where $m \times n = AC$ and $m + n = b$ 3. Rewrite: $ax^2 + mx + nx + c$ 4. Factor by grouping

Sign Rules for Factor Pairs

AC > 0, b > 0 \Rightarrow +, +

AC > 0, b < 0: both negative AC < 0: opposite signs