Polynomial Factoring Calculator
Enter Your Polynomial
Enter a polynomial to factor (e.g., x² - 5x + 6)
Factoring Methods
1. Common Factor
Factor out the GCF first
$6x^2 + 9x = 3x(2x + 3)$
2. Difference of Squares
$a^2 - b^2 = (a+b)(a-b)$
$x^2 - 9 = (x+3)(x-3)$
3. Perfect Square Trinomial
$a^2 ± 2ab + b^2 = (a±b)^2$
$x^2 + 6x + 9 = (x+3)^2$
4. Quadratic Trinomial
$ax^2 + bx + c$ → find factors
$x^2 + 5x + 6 = (x+2)(x+3)$
5. Grouping
Group terms and factor
$x^3 + x^2 + 2x + 2 = (x+1)(x^2+2)$
6. Sum/Difference of Cubes
$a^3 ± b^3 = (a±b)(a^2 ∓ ab + b^2)$
$x^3 + 8 = (x+2)(x^2-2x+4)$
Detailed Examples
Example 1: Simple Quadratic
Factor: $x^2 - 7x + 12$
Find two numbers that multiply to 12 and add to -7
Factors of 12: (1,12), (2,6), (3,4)
-3 × -4 = 12 and -3 + (-4) = -7 ✓
Answer: $(x - 3)(x - 4)$
Example 2: Difference of Squares
Factor: $4x^2 - 25$
Recognize: $4x^2 = (2x)^2$ and $25 = 5^2$
Apply formula: $a^2 - b^2 = (a+b)(a-b)$
Answer: $(2x + 5)(2x - 5)$
Example 3: Factor by Grouping
Factor: $2x^3 + 4x^2 + 3x + 6$
Group: $(2x^3 + 4x^2) + (3x + 6)$
Factor each group: $2x^2(x + 2) + 3(x + 2)$
Factor out $(x + 2)$: $(x + 2)(2x^2 + 3)$
Answer: $(x + 2)(2x^2 + 3)$
Special Factoring Patterns
Perfect Squares
- $x^2 + 2xy + y^2 = (x+y)^2$
- $x^2 - 2xy + y^2 = (x-y)^2$
Cubes
- $x^3 + y^3 = (x+y)(x^2-xy+y^2)$
- $x^3 - y^3 = (x-y)(x^2+xy+y^2)$
Practice Problems
- Factor: $x^2 + 8x + 15$
- Factor: $x^2 - 16$
- Factor: $2x^2 + 7x + 3$
- Factor: $x^3 - 27$
- Factor: $6x^2 - 13x + 6$
- Factor: $x^4 - 81$
Show Answers
- $(x + 3)(x + 5)$
- $(x + 4)(x - 4)$
- $(2x + 1)(x + 3)$
- $(x - 3)(x^2 + 3x + 9)$
- $(2x - 3)(3x - 2)$
- $(x^2 + 9)(x + 3)(x - 3)$