Finding Factors of a Number

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.

Factoring Out the Greatest Common Factor (GCF)

Factor: $6x + 12$

Identify the terms: Terms: $6x$ and $12$ = Two terms to factor

Find the GCF of coefficients: GCF of 6 and 12 is 6 = GCF = 6

Factor out the GCF: $6x + 12 = 6(x) + 6(2)$ = $6(x + 2)$

Verify by distributing: $6(x + 2) = 6x + 12$ ✓ = Correct!

Answer: $6x + 12 = 6(x + 2)$

Factoring with Variables

Factor: $x^2 + 5x$

Identify the terms: Terms: $x^2$ and $5x$ = Two terms with $x$

Find the GCF: Both terms contain $x$. GCF = $x$ = GCF = $x$

Factor out $x$: $x^2 + 5x = x(x) + x(5)$ = $x(x + 5)$

Verify by distributing: $x(x + 5) = x^2 + 5x$ ✓ = Correct!

Answer: $x^2 + 5x = x(x + 5)$

Factoring a Trinomial

Factor: $x^2 + 7x + 12$

Identify the pattern: This is a trinomial: $x^2 + bx + c$ where $b = 7$, $c = 12$ = Need two numbers

Find two numbers: Need numbers that multiply to 12 AND add to 7 = Product = 12, Sum = 7

List factor pairs of 12: (1, 12): $1 + 12 = 13$ ✗ (2, 6): $2 + 6 = 8$ ✗ (3, 4): $3 + 4 = 7$ ✓ = Numbers: 3 and 4

Write the factored form: $x^2 + 7x + 12 = (x + 3)(x + 4)$ = Factored!

Verify by FOIL: $(x+3)(x+4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ ✓ = Correct!

Answer: $x^2 + 7x + 12 = (x + 3)(x + 4)$

Factoring with Negative Terms

Factor: $x^2 - 2x - 15$

Identify the values: $b = -2$, $c = -15$ = Sum = -2, Product = -15

Analyze the signs: Product is negative, so one factor is positive, one is negative = Different signs

Find the numbers: Factors of 15: (1, 15), (3, 5) Try (3, -5): $3 + (-5) = -2$ ✓ Try (-3, 5): $-3 + 5 = 2$ ✗ = Numbers: 3 and -5

Write the factored form: $x^2 - 2x - 15 = (x + 3)(x - 5)$ = Factored!

Verify: $(x+3)(x-5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15$ ✓ = Correct!

Answer: $x^2 - 2x - 15 = (x + 3)(x - 5)$

Mistake: Forgetting to factor out the GCF first

Why: 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: $2(x^2 + 5x + 6) = 2(x+2)(x+3)$

Mistake: Getting the signs wrong in trinomials

Why: 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)

Mistake: Not verifying the answer

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

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

Mistake: Confusing factors with terms

Why: 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.

Area and Dimensions

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

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.

Breaking Down Complex Problems

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

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

Q: How do I know when an expression is completely factored?

A: 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.

Q: What if I cannot find two numbers that work?

A: Some trinomials cannot be factored using integers. For example, $x^2 + x + 1$ has no integer factors. These are called 'prime' or 'irreducible' polynomials. You'll learn other methods like the quadratic formula for these cases.

Q: Does the order of factors matter?

A: No! Because multiplication is commutative, $(x+2)(x+3)$ equals $(x+3)(x+2)$. Both are correct answers.