Powers and Polynomials 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.

Solved Examples

Jump directly to a worked example by factoring pattern.

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 $6x^3-18x^2+12x$.

Solution Steps
  1. Find the greatest common factor.
  2. Factor completely and state the result.
$$6x^3-18x^2+12x=6x(x^2-3x+2)$$ $$6x^3-18x^2+12x=6x(x-1)(x-2)$$

Grouping

Factor $x^3+2x^2+3x+6$.

Solution Steps
  1. Group the four terms into pairs and factor each pair.
  2. Factor out the common binomial and state the result.
$$x^3+2x^2+3x+6=x^2(x+2)+3(x+2)$$ $$x^3+2x^2+3x+6=(x+2)(x^2+3)$$

Difference of Squares

Factor $16x^2-81$.

Solution Steps
  1. Rewrite the expression as a difference of squares.
  2. Apply the conjugate-factor pattern.
$$16x^2-81=(4x)^2-9^2$$ $$16x^2-81=(4x-9)(4x+9)$$

Basic Trinomial

Factor $x^2-3x+2$.

Solution Steps
  1. Find two numbers whose product is $2$.
  2. Confirm their sum is $-3$.
  3. Write the factored trinomial.
$$(-1)(-2)=2$$ $$-1+(-2)=-3$$ $$x^2-3x+2=(x-1)(x-2)$$

Leading Coefficient

Factor $2x^2+5x+2$.

Solution Steps
  1. Multiply the leading and constant coefficients.
  2. Find numbers with product $4$ and sum $5$.
  3. Split the middle term.
  4. Group and factor each pair.
  5. State the factored expression.
$$a c=2\cdot 2=4$$ $$4+1=5,\quad 4\cdot 1=4$$ $$2x^2+5x+2=2x^2+4x+x+2$$ $$2x^2+5x+2=2x(x+2)+1(x+2)$$ $$2x^2+5x+2=(2x+1)(x+2)$$

Perfect Square Trinomial

Factor $x^2+6x+9$.

Solution Steps
  1. Recognize the first and last terms as squares.
  2. Check that the middle term is twice their product.
$$x^2+6x+9=x^2+2(x)(3)+3^2$$ $$x^2+6x+9=(x+3)^2$$

Sum of Cubes

Factor $8x^3+1$.

Solution Steps
  1. Rewrite the expression as a sum of cubes.
  2. Apply the sum-of-cubes pattern using SOAP signs.
  3. Multiply out the second factor and state the result.
$$8x^3+1=(2x)^3+1^3$$ $$8x^3+1=(2x+1)((2x)^2-(2x)(1)+1^2)$$ $$8x^3+1=(2x+1)(4x^2-2x+1)$$

Difference of Cubes

Factor $27x^3-8$.

Solution Steps
  1. Rewrite the expression as a difference of cubes.
  2. Apply the difference-of-cubes pattern.
  3. Multiply out the second factor and state the result.
$$27x^3-8=(3x)^3-2^3$$ $$27x^3-8=(3x-2)((3x)^2+(3x)(2)+2^2)$$ $$27x^3-8=(3x-2)(9x^2+6x+4)$$

Complete the Square

Complete the square for $x^2-6x$.

Solution Steps
  1. Take half of the $x$ coefficient.
  2. Square that value.
  3. Add the perfect-square term and write the result.
$$\frac{-6}{2}=-3$$ $$(-3)^2=9$$ $$x^2-6x+9=(x-3)^2$$