Polynomial Reference
Factoring Polynomials
Factoring rewrites a polynomial as a product. Start with structure, not guessing: remove common factors first, count terms, then choose the pattern that matches what remains.
Fact Table
| What You See | Try | Reminder |
|---|---|---|
| Every term shares a factor. | GCF first. | Never skip this step. |
| Four terms. | Grouping. | Pair terms so both pairs reveal the same binomial. |
| Three terms. | Trinomial factoring. | For $x^2+bx+c$, find numbers that multiply to $c$ and add to $b$. |
| Two terms with subtraction and squares. | Difference of squares. | Creates conjugates. |
| Two terms with cubes. | Sum or difference of cubes. | Use SOAP for the signs in the second factor. |
| Need vertex form or a perfect square. | Complete the square. | Take half of $b$, then square it. |
Content Formulas
GCF
$$ab+ac=a(b+c)$$
Difference of Squares
$$a^2-b^2=(a-b)(a+b)$$
Trinomial Pattern
$$x^2+bx+c=(x+m)(x+n)$$ $$mn=c,\quad m+n=b$$
Perfect Square Trinomials
$$a^2+2ab+b^2=(a+b)^2$$ $$a^2-2ab+b^2=(a-b)^2$$
Cubes
$$a^3+b^3=(a+b)(a^2-ab+b^2)$$ $$a^3-b^3=(a-b)(a^2+ab+b^2)$$
Completing the Square
$$x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2$$
SOAP is for cubes: Same sign in the binomial, Opposite sign next, Always Positive last.
Classic Examples
GCF First
Factor $6x^3-18x^2+12x$.
Factoring MovePull out the greatest common factor before checking any special pattern.
$$6x^3-18x^2+12x=6x(x^2-3x+2)$$
$$6x^3-18x^2+12x=6x(x-1)(x-2)$$
Grouping
Factor $x^3+3x^2+2x+6$.
Factoring MoveGroup four terms into pairs, then factor out the common binomial.
$$x^3+3x^2+2x+6=x^2(x+3)+2(x+3)$$
$$x^3+3x^2+2x+6=(x+3)(x^2+2)$$
Difference of Squares
Factor $16x^2-81$.
Factoring MoveTwo square terms with subtraction become conjugate factors.
$$16x^2-81=(4x)^2-9^2$$
$$16x^2-81=(4x-9)(4x+9)$$
Basic Trinomial
Factor $x^2-5x-24$.
Factoring MoveFind two numbers that multiply to $-24$ and add to $-5$.
$$(-8)(3)=-24$$
$$-8+3=-5$$
$$x^2-5x-24=(x-8)(x+3)$$
Leading Coefficient
Factor $6x^2+11x+3$.
Factoring MoveUse the $ac$ method: multiply $a\cdot c$, split the middle term, then group.
$$a c=6\cdot 3=18$$
$$9+2=11,\quad 9\cdot 2=18$$
$$6x^2+11x+3=6x^2+9x+2x+3$$
$$6x^2+11x+3=3x(2x+3)+1(2x+3)$$
$$6x^2+11x+3=(3x+1)(2x+3)$$
Perfect Square Trinomial
Factor $x^2+10x+25$.
Factoring MoveCheck whether the first and last terms are squares and the middle term is twice their product.
$$x^2+10x+25=x^2+2(x)(5)+5^2$$
$$x^2+10x+25=(x+5)^2$$
Sum of Cubes
Factor $8x^3+27$.
Factoring MoveUse SOAP: Same sign in the binomial, Opposite sign next, Always Positive last.
$$8x^3+27=(2x)^3+3^3$$
$$8x^3+27=(2x+3)((2x)^2-(2x)(3)+3^2)$$
$$8x^3+27=(2x+3)(4x^2-6x+9)$$
Difference of Cubes
Factor $125x^3-64$.
Factoring MoveThe binomial keeps the original minus sign; SOAP makes the trinomial signs plus, plus.
$$125x^3-64=(5x)^3-4^3$$
$$125x^3-64=(5x-4)((5x)^2+(5x)(4)+4^2)$$
$$125x^3-64=(5x-4)(25x^2+20x+16)$$
Complete the Square
Complete the square for $x^2-8x$.
Factoring MoveTake half of the $x$ coefficient, square it, then write the perfect square trinomial.
$$\frac{-8}{2}=-4$$
$$(-4)^2=16$$
$$x^2-8x+16=(x-4)^2$$