7.6Factoring Trinomials and Special Products
Factor trinomials using factor pairs and the AC method. Recognize and reverse special product patterns. Always check for a GCF first, and always verify by multiplying.
Why This Matters
Factoring trinomials and recognizing special products are essential skills for solving quadratic equations, simplifying rational expressions, and working with polynomial functions throughout Algebra 2, Precalculus, and Calculus. The AC method you learn here is the same technique used in Unit 8 to solve quadratic equations by factoring.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Quick Answer
How Do You Choose a Factoring Method?
First factor out the GCF. Then count the terms and look for a pattern. For x² + bx + c, find two numbers whose product is c and whose sum is b. For ax² + bx + c, use the AC method and grouping. For two terms, check for a difference of squares. For three terms, check for a perfect square trinomial.
Learning Goals
- Always check for a GCF before applying any other factoring method.
- Factor monic trinomials x² + bx + c using factor pairs.
- Use sign patterns to determine the signs of the factor pairs.
- Factor non-monic trinomials ax² + bx + c (a ≠ 1) using the AC method.
- Split the middle term and factor by grouping.
- Recognize and factor difference of squares: a² − b² = (a − b)(a + b).
- Recognize and factor perfect square trinomials.
- Completely factor expressions that require multiple steps.
- Identify prime polynomials that cannot be factored over the integers.
- Check every answer by multiplying the factors.
Key Vocabulary
Monic Trinomial
A trinomial with leading coefficient 1: x² + bx + c. Factor by finding two numbers whose product is c and sum is b.
Non-Monic Trinomial
A trinomial with leading coefficient a ≠ 1: ax² + bx + c. Use the AC method to factor.
AC Method
A factoring technique for ax² + bx + c: multiply a × c, find two numbers with that product and sum b, split the middle term, then factor by grouping.
Factoring by Grouping
Splitting a four-term polynomial into two pairs, factoring each pair, then factoring out the common binomial factor.
Difference of Squares
a² − b² = (a − b)(a + b). Requires subtraction and both terms must be perfect squares.
Perfect Square Trinomial
a² ± 2ab + b² = (a ± b)². First and last terms are perfect squares; middle term equals ±2ab.
Sum of Squares
a² + b² — does NOT factor over the real numbers. It is prime.
Completely Factored
An expression where no factor can be factored further over the integers.
Prime Polynomial
A polynomial that cannot be factored over the integers. Example: x² + 4.
Factor by Grouping
Group terms into pairs, factor each pair, then factor out the common binomial.
Factoring Strategy Flowchart
Factoring Strategy — Decision Flowchart
Is there a GCF?
Yes → Factor it out first. Always do this before anything else.
How many terms remain?
Count the terms in the remaining polynomial.
Two terms?
Check for difference of squares: a² − b² = (a − b)(a + b). Note: a sum of squares a² + b² does NOT factor over the reals.
Three terms?
Check for perfect square trinomial first. If not, use factor pairs (x² + bx + c) or the AC method (ax² + bx + c, a ≠ 1).
Four terms?
Try factoring by grouping: split into two pairs, factor each pair, then factor out the common binomial.
Can any factor be factored again?
Check every factor. An expression is completely factored only when no factor can be factored further.
Check by multiplication
Multiply all factors together. You must get the original expression.
Trinomial Sign Guide — x² + bx + c
Trinomial Sign Guide — x² + bx + c
Find two numbers p and q such that p · q = c and p + q = b. The signs of c and b tell you the signs of p and q.
| Sign of c | Sign of b | Signs of p and q | Example |
|---|---|---|---|
| + | + | Both positive | x² + 7x + 12 → p=3, q=4 |
| + | − | Both negative | x² − 5x + 6 → p=−2, q=−3 |
| − | + | One +, one − (larger is +) | x² + 2x − 15 → p=5, q=−3 |
| − | − | One +, one − (larger is −) | x² − 3x − 10 → p=−5, q=2 |
AC Method — ax² + bx + c (a ≠ 1)
AC Method — Step-by-Step Organizer
Use the AC method to factor ax² + bx + c when a ≠ 1. This method always works for factorable trinomials.
Check for a GCF
Factor out any GCF before applying the AC method.
Identify a, b, c
Write the trinomial in standard form ax² + bx + c.
Compute a × c
Multiply the leading coefficient a by the constant c.
Find two numbers
Find two integers whose product is ac and whose sum is b.
Split the middle term
Replace bx with the two numbers found in Step 4.
Factor by grouping
Group the four terms into two pairs. Factor each pair. Factor out the common binomial.
Check by multiplication
Multiply the two binomial factors. You must get the original trinomial.
Worked trace: 6x² + 11x + 3
a=6, b=11, c=3 → ac = 18
Find p, q: p·q=18, p+q=11 → p=9, q=2
Split: 6x² + 9x + 2x + 3
Group: (6x² + 9x) + (2x + 3)
Factor pairs: 3x(2x + 3) + 1(2x + 3)
= (3x + 1)(2x + 3)
Special Products — Factoring Reference
Special Products — Factoring Reference
Difference of Squares
a² − b² = (a − b)(a + b)
- Both terms must be perfect squares.
- Must be subtraction (difference), not addition.
- a² + b² does NOT factor over the real numbers.
Example: x² − 49 = (x − 7)(x + 7)
Perfect Square Trinomial (sum)
a² + 2ab + b² = (a + b)²
- First and last terms must be perfect squares.
- Middle term must equal +2ab.
- Check: 2·√(first)·√(last) = middle term.
Example: x² + 6x + 9 = (x + 3)²
Perfect Square Trinomial (difference)
a² − 2ab + b² = (a − b)²
- First and last terms must be perfect squares.
- Middle term must equal −2ab.
- Check: −2·√(first)·√(last) = middle term.
Example: x² − 10x + 25 = (x − 5)²
Worked Examples
Factor: x² + 7x + 12 (monic trinomial, c positive, b positive)
Check for GCF: GCF(1, 7, 12) = 1 — no GCF to factor out.
c = 12 > 0 and b = 7 > 0 → both numbers are positive.
Find p, q: p · q = 12 and p + q = 7.
Pairs: 1×12=12 (sum 13); 2×6=12 (sum 8); 3×4=12 (sum 7) ✓
p = 3, q = 4
x² + 7x + 12 = (x + 3)(x + 4)
Check: (x+3)(x+4) = x²+4x+3x+12 = x²+7x+12 ✓
Factor: x² − 5x + 6 (monic trinomial, c positive, b negative)
Check for GCF: none.
c = 6 > 0 and b = −5 < 0 → both numbers are negative.
Find p, q: p · q = 6 and p + q = −5.
Pairs: −1×−6=6 (sum −7); −2×−3=6 (sum −5) ✓
p = −2, q = −3
x² − 5x + 6 = (x − 2)(x − 3)
Check: (x−2)(x−3) = x²−3x−2x+6 = x²−5x+6 ✓
Factor: x² − 3x − 10 (monic trinomial, c negative)
Check for GCF: none.
c = −10 < 0 → one number is positive, one is negative.
b = −3 < 0 → the number with larger absolute value is negative.
Find p, q: p · q = −10 and p + q = −3.
Pairs: −5×2=−10 (sum −3) ✓
p = −5, q = 2
x² − 3x − 10 = (x − 5)(x + 2)
Check: (x−5)(x+2) = x²+2x−5x−10 = x²−3x−10 ✓
Factor: x² + 6x + 9 (perfect square trinomial)
Check for GCF: none.
Check for perfect square trinomial: first term x² = (x)²; last term 9 = (3)².
Middle term check: 2 · x · 3 = 6x ✓
Pattern: a² + 2ab + b² = (a + b)²; a = x, b = 3.
x² + 6x + 9 = (x + 3)²
Check: (x+3)² = x²+6x+9 ✓
Factor: x² − 49 (difference of squares)
Check for GCF: none.
Two terms, subtraction → check for difference of squares.
x² = (x)²; 49 = (7)² → both are perfect squares.
Pattern: a² − b² = (a − b)(a + b); a = x, b = 7.
x² − 49 = (x − 7)(x + 7)
Check: (x−7)(x+7) = x²+7x−7x−49 = x²−49 ✓
Factor: 6x² + 11x + 3 (AC method, a ≠ 1, c positive)
Check for GCF: GCF(6, 11, 3) = 1 — none.
a = 6, b = 11, c = 3 → ac = 6 × 3 = 18.
Find p, q: p · q = 18 and p + q = 11.
Pairs: 9 × 2 = 18 (sum 11) ✓
Split middle term: 6x² + 9x + 2x + 3
Group: (6x² + 9x) + (2x + 3)
Factor each pair: 3x(2x + 3) + 1(2x + 3)
Factor out common binomial: (3x + 1)(2x + 3)
Check: (3x+1)(2x+3) = 6x²+9x+2x+3 = 6x²+11x+3 ✓
Factor: 8x² − 2x − 3 (AC method, a ≠ 1, c negative)
Check for GCF: GCF(8, 2, 3) = 1 — none.
a = 8, b = −2, c = −3 → ac = 8 × (−3) = −24.
Find p, q: p · q = −24 and p + q = −2.
Pairs: −6 × 4 = −24 (sum −2) ✓
Split middle term: 8x² − 6x + 4x − 3
Group: (8x² − 6x) + (4x − 3)
Factor each pair: 2x(4x − 3) + 1(4x − 3)
Factor out common binomial: (2x + 1)(4x − 3)
Check: (2x+1)(4x−3) = 8x²−6x+4x−3 = 8x²−2x−3 ✓
Factor completely: 2x³ + 10x² + 12x (GCF first, then trinomial)
Check for GCF: GCF(2, 10, 12) = 2; GCF(x³, x², x) = x → GCF = 2x.
2x³ + 10x² + 12x = 2x(x² + 5x + 6)
Factor x² + 5x + 6: find p, q with p·q=6 and p+q=5 → p=2, q=3.
2x(x² + 5x + 6) = 2x(x + 2)(x + 3)
Check: 2x(x+2)(x+3) = 2x(x²+5x+6) = 2x³+10x²+12x ✓
Factor completely: 3x³ − 27x (GCF first, then difference of squares)
Check for GCF: GCF(3, 27) = 3; GCF(x³, x) = x → GCF = 3x.
3x³ − 27x = 3x(x² − 9)
Factor x² − 9: difference of squares, a = x, b = 3.
3x(x² − 9) = 3x(x − 3)(x + 3)
Check: 3x(x−3)(x+3) = 3x(x²−9) = 3x³−27x ✓
Factor: x² + 9 (prime polynomial — does not factor over the integers)
Check for GCF: none.
Two terms, addition → check for difference of squares.
x² + 9 is a SUM of squares, not a difference.
A sum of squares does NOT factor over the real numbers.
Check for trinomial: only two terms — not a trinomial.
Conclusion: x² + 9 is prime over the integers.
Common Mistakes
Forgetting to check for a GCF first — jumping straight to trinomial factoring.
Always factor out the GCF before applying any other method. It simplifies every subsequent step.
Using numbers that multiply to c correctly but do not add to b — e.g., using 4 and 3 for x² + 8x + 12 (4+3=7, not 8).
Check both conditions: p · q = c AND p + q = b. Both must hold.
Incorrect sign choices — e.g., writing (x + 5)(x − 2) for x² − 3x − 10 instead of (x − 5)(x + 2).
Use the sign guide: c < 0 means one factor is positive and one is negative. The one with larger absolute value matches the sign of b.
Treating a sum of squares as factorable — e.g., writing x² + 9 = (x + 3)².
x² + 9 is prime over the real numbers. Only a DIFFERENCE of squares factors: a² − b² = (a − b)(a + b).
Confusing difference of squares with a perfect square trinomial — e.g., writing x² − 9 = (x − 3)².
(x − 3)² = x² − 6x + 9, which has a middle term. x² − 9 has no middle term — it is a difference of squares: (x − 3)(x + 3).
Skipping the middle-term split in the AC method — trying to factor directly without grouping.
The AC method requires splitting bx into two terms, then grouping. Do not skip this step.
Grouping terms incorrectly in the AC method — e.g., grouping the first and last terms instead of consecutive pairs.
After splitting the middle term, group the first two terms together and the last two terms together.
Stopping before complete factorization — e.g., leaving 2x(x² − 9) as the final answer.
After the GCF, always check whether the remaining factor can be factored further. 2x(x² − 9) = 2x(x − 3)(x + 3).
Failing to check by multiplication — assuming the factored form is correct.
Always multiply the factors together. If you do not get the original expression, find the error.
Assuming every trinomial factors over the integers — spending time on a prime polynomial.
If no integer pair satisfies both conditions, the polynomial is prime over the integers. State this clearly.
Factoring x² + bx + c using only the value of c — ignoring b.
Both conditions must hold: p · q = c AND p + q = b. A pair that gives the right product but wrong sum is incorrect.
Applying the monic trinomial method to ax² + bx + c when a ≠ 1.
When a ≠ 1, use the AC method. The monic method only works when the leading coefficient is 1.
Guided Practice
Guided Practice Video: Factoring Trinomials
Review trinomial factoring strategies and special product patterns before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Factor: x² + 9x + 20
Hint: c = 20 > 0, b = 9 > 0 → both numbers positive. Find p, q: p·q=20, p+q=9.
Factor: x² − 7x + 10
Hint: c = 10 > 0, b = −7 < 0 → both numbers negative. Find p, q: p·q=10, p+q=−7.
Factor: x² + 2x − 15
Hint: c = −15 < 0 → one positive, one negative. b = 2 > 0 → the positive number has larger absolute value.
Factor: x² − 16
Hint: Two terms, subtraction. Are both terms perfect squares? Apply the difference of squares pattern.
Factor: 2x² + 7x + 3 (AC method)
Hint: a=2, c=3 → ac=6. Find p, q: p·q=6, p+q=7. Split the middle term, then group.
Factor completely: 4x³ − 36x
Hint: Factor out the GCF first. Then check the remaining binomial for a special product pattern.
Math Tips
Always check for a GCF before applying any other factoring method.
For x² + bx + c: the sign of c tells you whether the factors have the same sign or opposite signs.
For the AC method: if you cannot find integer factors of ac that add to b, the trinomial is prime over the integers.
A sum of squares (a² + b²) is always prime over the real numbers.
After factoring, check that no remaining factor can be factored further.
The AC method works for all factorable trinomials — including monic ones (a = 1).
Perfect square trinomials: check that the middle term equals exactly ±2ab.
Always verify by multiplying the factors back together.
Interactive Practice — 3 Questions
Which is the correct factored form of x² − 8x + 15?
Which expression is prime over the integers?
What is the first step when factoring 6x² + 9x + 3?