Introduction to Factoring
Learn what factoring is and how to break down expressions into their component factors.
Definition
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Worked Examples
Find all factor pairs of 24.
Start with 1
→ Factor pair: (1, 24)
Try 2
, so → Factor pair: (2, 12)
Try 3
, so → Factor pair: (3, 8)
Try 4
, so → Factor pair: (4, 6)
Check 5
(not a whole number) → 5 is not a factor
Stop at square root
, 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 , first factor out 2:
Getting the signs wrong in trinomials
Why it's wrong: When is negative, the two factors have opposite signs. When is positive, they have the same sign.
Correct: (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: ✓
Confusing factors with terms
Why it's wrong: Factors multiply together; terms are added or subtracted. In , the terms are and , but is a factor of both.
Correct: Terms: parts separated by + or -. Factors: parts that multiply together.
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Click on the circle to change the fraction
Interactive Sandbox
Expression Calculator
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Practice Problems
17 problemsWhich of the following is a factor of 12?
Why It Matters
- Solving equations: To solve , we factor to
- 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
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 square meters. What are the dimensions? Factor: , so the dimensions are by meters.
A rectangular pool has an area of square feet.
What are the dimensions of the pool?
Step 1: Write the mathematical expression
Factor :
Breaking Down Complex Problems
In engineering and science, factoring helps simplify complex expressions to make calculations easier.
Example:
A physics formula might simplify from to by factoring.
Simplify the expression .
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 , find two numbers that multiply to and add to
- 4The signs of and tell you the signs of the factors
- 5Always verify your answer by multiplying the factors back together
Frequently Asked Questions
Glossary
- Factor
- A number or expression that divides evenly into another. In , 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 .
- Prime polynomial
- A polynomial that cannot be factored into polynomials with integer coefficients.