Introduction to Factoring

Learn what factoring is and how to break down expressions into their component factors.

Intermediate25 minLesson
Printable worksheet

Definition

Factoring is the process of breaking down a number or expression into smaller parts (called factors) that multiply together to give the original.
For numbers:
12=2×6=3×4=2×2×312 = 2 \times 6 = 3 \times 4 = 2 \times 2 \times 3
For algebraic expressions:
x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)
Factoring is like reverse multiplication. Instead of multiplying to get a product, we start with the product and find what was multiplied.

Try it now

Which of the following is a factor of 12?

Worked Examples

Find all factor pairs of 24.
1

Start with 1

24=1×2424 = 1 \times 24 → Factor pair: (1, 24)
2

Try 2

24÷2=1224 \div 2 = 12, so 24=2×1224 = 2 \times 12 → Factor pair: (2, 12)
3

Try 3

24÷3=824 \div 3 = 8, so 24=3×824 = 3 \times 8 → Factor pair: (3, 8)
4

Try 4

24÷4=624 \div 4 = 6, so 24=4×624 = 4 \times 6 → Factor pair: (4, 6)
5

Check 5

24÷5=4.824 \div 5 = 4.8 (not a whole number) → 5 is not a factor
6

Stop at square root

24≈4.9\sqrt{24} \approx 4.9, and we've checked up to 4 → All pairs found

Common Mistakes

Forgetting to factor out the GCF first

Why it's wrong: Always check for a GCF before trying other factoring methods. It makes the remaining factoring much easier.

Correct: For 2x2+10x+122x^2 + 10x + 12, first factor out 2: 2(x2+5x+6)=2(x+2)(x+3)2(x^2 + 5x + 6) = 2(x+2)(x+3)

Getting the signs wrong in trinomials

Why it's wrong: When cc is negative, the two factors have opposite signs. When cc is positive, they have the same sign.

Correct: x2−5x+6=(x−2)(x−3)x^2 - 5x + 6 = (x-2)(x-3) (both negative because sum is negative, product is positive)

Not verifying the answer

Why it's wrong: Always multiply your factors back together to check. It takes 30 seconds and catches errors.

Correct: After factoring, use FOIL or distribution to verify: (x+2)(x+3)=x2+5x+6(x+2)(x+3) = x^2 + 5x + 6 ✓

Confusing factors with terms

Why it's wrong: Factors multiply together; terms are added or subtracted. In 2x+42x + 4, the terms are 2x2x and 44, but 22 is a factor of both.

Correct: Terms: parts separated by + or -. Factors: parts that multiply together.

Interactive Visual

Factor Tree

Number:
Composite
Prime

Enter a number to see its factor tree and prime factorization.

Area model: factoring a trinomial

Arrange x², the x-strips and the unit squares into one rectangle: its sides are the factors.

x2x^2
3x3x
4x4x
1212
xx
33
xx
44

Length × width = area

x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x + 3)(x + 4)

Interactive Sandbox

Expression Calculator

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Practice Problems

17 problems
Problem 1 of 17
Easy
Which of the following is a factor of 12?

Why It Matters

Factoring is one of the most important skills in algebra:
  • Solving equations: To solve x2+5x+6=0x^2 + 5x + 6 = 0, we factor to (x+2)(x+3)=0(x+2)(x+3) = 0
  • Simplifying fractions: Factor to cancel common terms
  • Finding patterns: Prime factorization helps with GCF and LCM
  • Real-world applications: Breaking problems into smaller, manageable parts
Mastering factoring opens the door to advanced mathematics, including calculus and beyond!

Real World Applications

Area and Dimensions

When you know the area of a rectangle and need to find possible dimensions, you factor.

Example:

A garden has an area of x2+8x+15x^2 + 8x + 15 square meters. What are the dimensions? Factor: (x+3)(x+5)(x+3)(x+5), so the dimensions are (x+3)(x+3) by (x+5)(x+5) meters.

1Try It Yourself

A rectangular pool has an area of x2+10x+24x^2 + 10x + 24 square feet.

What are the dimensions of the pool?

Step 1: Write the mathematical expression

Factor x2+10x+24x^2 + 10x + 24:

Breaking Down Complex Problems

In engineering and science, factoring helps simplify complex expressions to make calculations easier.

Example:

A physics formula might simplify from 2t2+6t2t\frac{2t^2 + 6t}{2t} to 2t(t+3)2t=t+3\frac{2t(t + 3)}{2t} = t + 3 by factoring.

2Try It Yourself

Simplify the expression 3x2+9x3x\frac{3x^2 + 9x}{3x}.

What does this simplify to?

Step 1: Write the mathematical expression

First factor the numerator:

Key Takeaways

  • 1Factoring breaks down numbers or expressions into parts that multiply together
  • 2Always look for a Greatest Common Factor (GCF) first
  • 3For trinomials x2+bx+cx^2 + bx + c, find two numbers that multiply to cc and add to bb
  • 4The signs of bb and cc tell you the signs of the factors
  • 5Always verify your answer by multiplying the factors back together

Frequently Asked Questions

An expression is completely factored when you cannot factor any of the remaining factors further. For example, (x+2)(x+3)(x+2)(x+3) is complete, but 2(x2+5x+6)2(x^2+5x+6) is not because the trinomial can still be factored.
An expression is completely factored when you cannot factor any of the remaining factors further. For example, (x+2)(x+3)(x+2)(x+3) is complete, but 2(x2+5x+6)2(x^2+5x+6) is not because the trinomial can still be factored.
Some trinomials cannot be factored using integers. For example, x2+x+1x^2 + x + 1 has no integer factors. These are called 'prime' or 'irreducible' polynomials. You'll learn other methods like the quadratic formula for these cases.
No! Because multiplication is commutative, (x+2)(x+3)(x+2)(x+3) equals (x+3)(x+2)(x+3)(x+2). Both are correct answers.

Glossary

Factor
A number or expression that divides evenly into another. In 12=3×412 = 3 \times 4, both 3 and 4 are factors of 12.
Factoring
The process of rewriting a number or expression as a product of its factors.
Greatest Common Factor (GCF)
The largest factor shared by two or more numbers or terms.
Trinomial
A polynomial with exactly three terms, like x2+5x+6x^2 + 5x + 6.
Prime polynomial
A polynomial that cannot be factored into polynomials with integer coefficients.
Today's challengeOne problem, three tries

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