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Factoring Trinomials Calculator

Factor trinomials of the form ax² + bx + c into binomial products. Get instant factored forms with step-by-step solutions, discriminant analysis, and verification for algebra homework and test prep.

Factor Your Trinomial

Enter coefficients for ax² + bx + c

Trinomial: ax² + bx + c
Your trinomial:
x² + 5x + 6

Factoring Result

✓ Factored Form:
(x + 2)(x + 3)
Discriminant (b² - 4ac)
1
Two real roots
Method Used
AC Method

Step-by-Step Solution

Original trinomial: x² + 5x + 6
Step 1: GCF = 1 (no common factor)
Step 2: Calculate discriminant: b² - 4ac = 5² - 4(1)(6) = 1
Step 3: Use AC method to factor
Find two numbers that multiply to ac = 1 × 6 = 6 and add to b = 5
Found: 2 and 3 (2 × 3 = 6, 2 + 3 = 5)
Step 4: Rewrite middle term: x² + 5x + 6 = x² + 2x + 3x + 6
Step 5: Factor by grouping
Final factored form: (x + 2)(x + 3)

✓ Verification

Expand (x + 2)(x + 3) to get x² + 5x + 6

What is Factoring Trinomials?

Factoring trinomials is the process of breaking down a quadratic expression with three terms (ax² + bx + c) into a product of two binomials. This reverse operation of expanding is fundamental in algebra and essential for solving quadratic equations.

Why Factor Trinomials?

  • ✓Solve equations: Finding x-intercepts and zeros of quadratic functions
  • ✓Simplify expressions: Reducing complex algebraic fractions
  • ✓Analyze functions: Understanding behavior of parabolas
  • ✓Build foundations: Essential skill for advanced mathematics

How to Factor Trinomials

Factoring Methods

Method 1: Simple Factoring (when a = 1)

x² + bx + c = (x + m)(x + n)

Find m and n where: m × n = c and m + n = b

Method 2: AC Method (when a ≠ 1)

ax² + bx + c

1. Multiply a × c
2. Find factors of ac that add to b
3. Split middle term and group

Step-by-Step Process:

  1. 1
    Check for GCF (Greatest Common Factor)
    Factor out any common factors from all terms first
  2. 2
    Calculate discriminant (b² - 4ac)
    Determines if trinomial is factorable over integers
  3. 3
    Find factor pairs
    Use appropriate method based on coefficient a
  4. 4
    Write factored form
    Express as product of two binomials and verify

Important Considerations

⚠️ Common Mistakes

Always check your answer by expanding. Look for GCF first. Not all trinomials are factorable over integers.

🔍 Check the Discriminant

b² - 4ac tells factorability

  • • Perfect square → factorable
  • • Negative → not factorable (real)
  • • Not perfect square → irrational
✓ Always Verify

Expand to check your answer

  • • Use FOIL method
  • • Compare with original
  • • Check all signs carefully
📝 Sign Patterns

Signs guide your factoring

  • • + + + → both factors positive
  • • + - + → both factors negative
  • • + - - or + + - → mixed signs
⚡ Special Cases

Recognize patterns quickly

  • • Perfect square: (a ± b)²
  • • Difference of squares: a² - b²
  • • Prime trinomials (unfactorable)

Factoring Methods Comparison

MethodWhen to UseExample
Simple MethodWhen a = 1x² + 5x + 6 = (x + 2)(x + 3)
AC MethodWhen a ≠ 12x² + 7x + 3 = (2x + 1)(x + 3)
Perfect Squareb² = 4acx² + 6x + 9 = (x + 3)²
Difference of Squaresb = 0, c < 0x² - 16 = (x + 4)(x - 4)

Example Cases

Case 1: Simple Trinomial (a = 1)

Problem: x² + 5x + 6
Find: Two numbers that multiply to 6 and add to 5
Numbers: 2 and 3 (2 × 3 = 6, 2 + 3 = 5)
Answer: (x + 2)(x + 3)
Verify: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓

Case 2: AC Method (a ≠ 1)

Problem: 2x² + 7x + 3
AC = 2 × 3 = 6
Find factors of 6 that add to 7: 6 and 1
Rewrite: 2x² + 6x + 1x + 3
Group: 2x(x + 3) + 1(x + 3)
Answer: (2x + 1)(x + 3)

Case 3: Perfect Square Trinomial

Problem: x² + 10x + 25
Pattern: a² + 2ab + b²
Recognize: 25 = 5², and 10x = 2(x)(5)
Answer: (x + 5)²
This is faster than the general method!

Common Patterns to Memorize

Perfect Square Trinomials

a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²

Check if first and last terms are perfect squares, and middle term = 2√(first × last)

Difference of Squares

a² - b² = (a + b)(a - b)

Only works with subtraction. Both terms must be perfect squares.

Sign Patterns

• x² + bx + c: Same signs in factors
• x² - bx + c: Both negative
• x² + bx - c: Different signs
• x² - bx - c: Different signs

Quick Checks

✓ GCF first, always
✓ Calculate discriminant
✓ Look for special patterns
✓ Verify by expanding

Frequently Asked Questions

What is a trinomial in algebra?
A trinomial is a polynomial with exactly three terms. The most common form is ax² + bx + c, where a, b, and c are constants and a ≠ 0. Trinomials are quadratic expressions that can often be factored into two binomials, such as (x + m)(x + n).
How do I factor trinomials with a = 1?
When a = 1 (x² + bx + c), find two numbers that multiply to c and add to b. For example, x² + 5x + 6 factors to (x + 2)(x + 3) because 2 × 3 = 6 and 2 + 3 = 5. This is the simplest factoring case and forms the foundation for more complex trinomials.
What is the AC method for factoring?
The AC method works for any trinomial ax² + bx + c. Multiply a × c, then find two numbers that multiply to ac and add to b. Rewrite the middle term using these numbers, then factor by grouping. This method is particularly useful when a ≠ 1 and ensures you find all integer factorizations.
How do I know if a trinomial is factorable?
Calculate the discriminant b² - 4ac. If it's a perfect square (like 0, 1, 4, 9, 16...), the trinomial is factorable over integers. If negative, no real factors exist. If positive but not a perfect square, factors exist but may involve irrational numbers or require the quadratic formula.
What are special factoring patterns for trinomials?
Special patterns include: (1) Perfect square trinomials: a² + 2ab + b² = (a + b)², (2) Difference of squares: a² - b² = (a + b)(a - b), (3) Sum/difference of cubes, and (4) Grouping methods. Recognizing these patterns speeds up factoring significantly.
Why is factoring trinomials important?
Factoring is essential for solving quadratic equations, simplifying algebraic expressions, finding zeros of functions, and analyzing parabolas. It's a fundamental skill in algebra that applies to calculus, physics, engineering, and computer science. Mastering trinomial factoring builds problem-solving abilities for more advanced mathematics.

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