Introduction to Factoring

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

Intermediate25 minLesson

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 \times 6 = 3 \times 4 = 2 \times 2 \times 3

For algebraic expressions:

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.

Start with 1

$24 = 1 \times 24$ → Factor pair: (1, 24)

Try 2

$24 \div 2 = 12$ , so $24 = 2 \times 12$ → Factor pair: (2, 12)

Try 3

$24 \div 3 = 8$ , so $24 = 3 \times 8$ → Factor pair: (3, 8)

Try 4

$24 \div 4 = 6$ , so $24 = 4 \times 6$ → Factor pair: (4, 6)

Check 5

$24 \div 5 = 4.8$ (not a whole number) → 5 is not a factor

Stop at square root

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

Answer: Factor pairs of 24: (1, 24), (2, 12), (3, 8), (4, 6). All factors: 1, 2, 3, 4, 6, 8, 12, 24.

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 $2x^2 + 10x + 12$ , first factor out 2: $2(x^2 + 5x + 6) = 2(x+2)(x+3)$

Getting the signs wrong in trinomials

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

Correct: $x^2 - 5x + 6 = (x-2)(x-3)$ (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: $(x+2)(x+3) = x^2 + 5x + 6$ ✓

Confusing factors with terms

Why it's wrong: Factors multiply together; terms are added or subtracted. In $2x + 4$ , the terms are $2x$ and $4$ , but $2$ 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: distributive property

a rows of (x + b): the area is a·x plus a·b.

a4b3

Length × width = area

$4(x + 3) = 4x + 12$

Interactive Sandbox

Expression Calculator

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History

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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 $x^2 + 5x + 6 = 0$ , we factor to $(x+2)(x+3) = 0$
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 $x^2 + 8x + 15$ square meters. What are the dimensions? Factor: $(x+3)(x+5)$ , so the dimensions are $(x+3)$ by $(x+5)$ meters.

1Try It Yourself

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

What are the dimensions of the pool?

Step 1: Write the mathematical expression

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

Breaking Down Complex Problems

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

Example:

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

2Try It Yourself

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

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 $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$
4The signs of $b$ and $c$ 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)

is complete, but

2(x^2+5x+6)

is not because the trinomial can still be factored.

Glossary

Factor: A number or expression that divides evenly into another. In $12 = 3 \times 4$ , 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 $x^2 + 5x + 6$ .
Prime polynomial: A polynomial that cannot be factored into polynomials with integer coefficients.