Factoring Trinomials (General)

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

Advanced25 min16 problems
Practice Problems

Definition

When factoring a trinomial ax2+bx+cax^2 + bx + c where a≠1a \neq 1, we use the AC method (also called the grouping method).

The AC Method

Step 1: Multiply a×ca \times c to get the "AC product"
Step 2: Find two numbers that:
  • Multiply to give AC
  • Add to give bb
Step 3: Rewrite the middle term using these two numbers
Step 4: Factor by grouping
ax2+bx+c=(px+q)(rx+s)ax^2 + bx + c = (px + q)(rx + s)
where p×r=ap \times r = a, q×s=cq \times s = c and ps+qr=bps + qr = b

Try it now

What is the AC product for 2x2+5x+32x^2 + 5x + 3?

Worked Examples

Factor 2x2+7x+32x^2 + 7x + 3
1

Identify a, b, and c

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

Calculate AC product

AC=2×3=6AC = 2 \times 3 = 6 → AC=6AC = 6
3

Find two numbers that multiply to 6 and add to 7

Factors of 6: (1,6)(1, 6), (2,3)(2, 3) 1+6=71 + 6 = 7 ✓ → Numbers: 11 and 66
4

Rewrite middle term

2x2+1x+6x+32x^2 + 1x + 6x + 3 → Split 7x7x into 1x+6x1x + 6x
5

Group and factor

(2x2+1x)+(6x+3)(2x^2 + 1x) + (6x + 3) =x(2x+1)+3(2x+1)= x(2x + 1) + 3(2x + 1) → Common factor: (2x+1)(2x + 1)
6

Factor out common binomial

(2x+1)(x+3)(2x + 1)(x + 3) → Factored form

Common Mistakes

Forgetting to multiply a×ca \times c and just using cc

Why it's wrong: When a≠1a \neq 1, the product AC is different from just cc. For 2x2+7x+32x^2 + 7x + 3, AC = 6, not 3.

Correct: Always calculate AC=a×cAC = 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 bb. 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.

Interactive Visual

Function graph

-5-4-3-2-1012-5051015xy(-3, 0)(-0.5, 0)(-1.75, -3.13)

Tap or hover over the graph to read off values.

2x^2 + 7x + 3(2x + 1)(x + 3)
  • Zero (x-intercept): (-3, 0)
  • Zero (x-intercept): (-0.5, 0)
  • Zero (x-intercept): (-3, 0)
  • Zero (x-intercept): (-0.5, 0)
  • Vertex: (-1.75, -3.13)
  • Vertex: (-1.75, -3.13)

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the AC product for 2x2+5x+32x^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 2x2+7x+3=02x^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:

A ball is thrown upward from a 15 m high roof. After tt seconds its height is h=−5t2+10t+15h = -5t^2 + 10t + 15 meters. Factoring gives −5(t2−2t−3)=−5(t−3)(t+1)-5(t^2 - 2t - 3) = -5(t-3)(t+1), so the ball hits the ground at t=3t = 3 seconds.

1Try It Yourself

A ball is thrown upward from a height of 2 m. After tt seconds its height is h=−5t2+9t+2h = -5t^2 + 9t + 2 meters.

After how many seconds does the ball hit the ground?

Step 1: Write the mathematical expression

Set h=0h = 0 and factor. Time in seconds:

Business Profit Analysis

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

Example:

A company's profit is P=−2x2+14x−24P = -2x^2 + 14x - 24 thousand dollars, where xx is units in hundreds. Factoring as −2(x−3)(x−4)-2(x-3)(x-4) shows break-even at 300 and 400 units.

2Try It Yourself

A workshop's profit is P=−4x2+26x−30P = -4x^2 + 26x - 30 thousand dollars, where xx is the number of items made, in hundreds.

Which expression is the complete factorization of PP?

Step 1: Choose the right answer

Factor out −2-2 first, then use the AC method on the trinomial that is left.

Key Takeaways

  • 1The AC method factors trinomials ax2+bx+cax^2 + bx + c when a≠1a \neq 1
  • 2Calculate AC=a×cAC = a \times c, then find two numbers that multiply to AC and add to bb
  • 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 bb, the trinomial cannot be factored over the integers (it is prime). You can still find its roots with the quadratic formula.
If no integer pair multiplies to AC and adds to bb, the trinomial cannot be factored over the integers (it is prime). You can still find its roots with the quadratic formula.
You can split the middle term in either order (e.g., −2x+12x-2x + 12x or 12x−2x12x - 2x), as long as you group correctly to get the same binomial factor.
Always check if all three coefficients share a common factor. Factoring out the GCF first makes the remaining trinomial simpler to factor.

Glossary

AC Method
A factoring technique where you multiply a×ca \times c, find a factor pair, and use grouping
Leading Coefficient
The coefficient aa in ax2+bx+cax^2 + bx + c, the number in front of x2x^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(px + q)^2, like 4x2−12x+9=(2x−3)24x^2 - 12x + 9 = (2x-3)^2

Formula Card

AC Method Steps

ax2+bx+cax^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⇒+,+AC > 0, b > 0 \Rightarrow +, +

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

Today's challengeOne problem, three tries

More in This Topic

Introduction to Factoring

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Factoring Out the GCF

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Factoring by Grouping

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Factoring Trinomials (a=1)

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