Factoring Polynomials

Learn GCF, trinomials, and special patterns

Factoring Polynomials

Greatest Common Factor (GCF)

Factor out the largest common expression.

Example: 6x3+9x2=3x2(2x+3)6x^3 + 9x^2 = 3x^2(2x + 3)

Factoring Trinomials

For x2+bx+cx^2 + bx + c, find two numbers that multiply to cc and add to bb.

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

Difference of Squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Example: x2−25=(x+5)(x−5)x^2 - 25 = (x + 5)(x - 5)

Perfect Square Trinomials

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

Example: x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2

📚 Practice Problems

1Problem 1easy

❓ Question:

Factor completely: 6x2+9x6x^2 + 9x

💡 Show Solution

Step 1: Find the GCF of the terms

  • GCF of coefficients: 66 and 99 → GCF = 33
  • GCF of variables: x2x^2 and xx → GCF = xx
  • Overall GCF: 3x3x

Step 2: Factor out 3x3x 6x2+9x=3x(2x+3)6x^2 + 9x = 3x(2x + 3)

Check: 3x(2x+3)=6x2+9x3x(2x + 3) = 6x^2 + 9x ✓

Answer: 3x(2x+3)3x(2x + 3)

2Problem 2medium

❓ Question:

Factor: x2−9x+20x^2 - 9x + 20

💡 Show Solution

We need two numbers that multiply to 20 and add to -9

List factor pairs of 20:

  • 1×20=201 \times 20 = 20, sum = 2121 ✗
  • 2×10=202 \times 10 = 20, sum = 1212 ✗
  • 4×5=204 \times 5 = 20, sum = 99 ✗
  • (−4)×(−5)=20(-4) \times (-5) = 20, sum = −9-9 ✓

Answer: (x−4)(x−5)(x - 4)(x - 5)

Check: (x−4)(x−5)=x2−5x−4x+20=x2−9x+20(x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20 ✓

3Problem 3hard

❓ Question:

Factor completely: 3x3−12x3x^3 - 12x

💡 Show Solution

Step 1: Factor out the GCF (3x3x) 3x3−12x=3x(x2−4)3x^3 - 12x = 3x(x^2 - 4)

Step 2: Notice x2−4x^2 - 4 is a difference of squares x2−4=x2−22=(x+2)(x−2)x^2 - 4 = x^2 - 2^2 = (x + 2)(x - 2)

Step 3: Combine all factors 3x3−12x=3x(x+2)(x−2)3x^3 - 12x = 3x(x + 2)(x - 2)

Answer: 3x(x+2)(x−2)3x(x + 2)(x - 2)

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