Factoring Decision Flowchart
Follow the steps in order every time you factor a polynomial.
then continue with remaining factor
\(a^2 - b^2\)
\(x^2+bx+c\)
that multiply to \(c\)
and add to \(b\)
Trial and Error
Group 2+2,
factor each group
Always check your answer!
Multiply your factors back out using FOIL or distribution. You should get the original polynomial.
Greatest Common Factor (GCF)
What is GCF?
The GCF is the largest factor (number and/or variable) that divides evenly into every term of the polynomial. Always factor out the GCF first before applying any other method.
Steps to Factor out the GCF
- Find the GCF of all the coefficients (the numerical GCF).
- Find the lowest power of each variable that appears in all terms.
- Write the GCF in front of parentheses.
- Divide each term by the GCF to get what goes inside the parentheses.
Example 1 — Numerical GCF only
Factor \(12x^3 - 8x^2 + 4x\).
GCF of 12, 8, 4 is 4. Lowest power of \(x\) in all terms: \(x^1\). So GCF = \(4x\).
\(12x^3 - 8x^2 + 4x = 4x(3x^2 - 2x + 1)\)
Example 2 — Variable GCF
Factor \(15x^4y^3 - 10x^2y^5 + 5x^3y^2\).
GCF of coefficients: 5. Lowest power of \(x\): \(x^2\). Lowest power of \(y\): \(y^2\). GCF = \(5x^2y^2\).
\(= 5x^2y^2(3x^2y - 2y^3 + x)\)
Example 3 — GCF is a binomial
Factor \(x(x+3) + 2(x+3)\).
GCF is the binomial \((x+3)\).
\(= (x+3)(x+2)\)
Practice Problems
1. Factor \(6x^2 + 9x\).
\(= 3x(2x + 3)\)
2. Factor \(14a^3b^2 - 21a^2b^3 + 7ab\).
\(= 7ab(2a^2b - 3ab^2 + 1)\)
3. Factor \(3x(x-5) - 7(x-5)\).
\(= (x-5)(3x-7)\)
4. Factor \(20x^4 - 16x^3 + 8x^2\).
\(= 4x^2(5x^2 - 4x + 2)\)
Difference of Squares
Formula
\[ a^2 - b^2 = (a + b)(a - b) \]Applies when you have two perfect square terms separated by a minus sign.
Note: A sum of squares \(a^2 + b^2\) does not factor over the reals.
Recognizing Perfect Squares
Numbers: \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots\)
Variables: \(x^2, x^4, x^6, \ldots\) (any even exponent)
Example 1
Factor \(x^2 - 49\).
\(x^2 = (x)^2\), \(49 = (7)^2\)
\(= (x+7)(x-7)\)
Example 2
Factor \(9x^2 - 25\).
\(9x^2 = (3x)^2\), \(25 = (5)^2\)
\(= (3x+5)(3x-5)\)
Example 3 — Factor completely
Factor \(x^4 - 16\).
\(= (x^2+4)(x^2-4)\) — but \(x^2-4\) factors again!
\(= (x^2+4)(x+2)(x-2)\)
Example 4 — GCF first
Factor \(2x^2 - 18\).
GCF = 2: \(2(x^2-9) = 2(x+3)(x-3)\)
Practice Problems
1. Factor \(x^2 - 64\).
2. Factor \(4x^2 - 9\).
3. Factor \(16a^2 - 25b^2\).
4. Factor completely: \(3x^3 - 75x\).
Factoring Trinomials
Case 1: Leading Coefficient = 1 (\(x^2 + bx + c\))
Method: Find two numbers
Find two integers \(p\) and \(q\) such that:
\[ p \cdot q = c \quad \text{and} \quad p + q = b \]Then: \(x^2 + bx + c = (x + p)(x + q)\)
Example 1
Factor \(x^2 + 7x + 12\).
Need: \(p \cdot q = 12\) and \(p + q = 7\).
Try: \(3 \times 4 = 12\) and \(3 + 4 = 7\). ✓
\(= (x+3)(x+4)\)
Example 2 — Negative constant
Factor \(x^2 - 2x - 15\).
Need: \(p \cdot q = -15\) and \(p + q = -2\).
Try: \(-5 \times 3 = -15\) and \(-5 + 3 = -2\). ✓
\(= (x-5)(x+3)\)
Case 2: Leading Coefficient ≠ 1 (\(ax^2 + bx + c\))
AC Method (Splitting the Middle)
- Multiply \(a \cdot c\) (the product \(AC\)).
- Find two numbers that multiply to \(AC\) and add to \(b\).
- Rewrite the middle term using those two numbers.
- Factor by grouping.
Example 3 — AC Method
Factor \(2x^2 + 7x + 3\).
\(AC = 2 \times 3 = 6\). Need two numbers that multiply to 6 and add to 7: \(1\) and \(6\).
Rewrite: \(2x^2 + 1x + 6x + 3\)
Group: \(x(2x+1) + 3(2x+1) = (2x+1)(x+3)\)
Example 4 — Negative middle
Factor \(6x^2 - 11x + 4\).
\(AC = 24\). Need two numbers: multiply to 24, add to \(-11\): \(-3\) and \(-8\).
Rewrite: \(6x^2 - 3x - 8x + 4\)
Group: \(3x(2x-1) - 4(2x-1) = (2x-1)(3x-4)\)
Practice Problems
1. Factor \(x^2 + 8x + 15\).
\(=(x+3)(x+5)\)
2. Factor \(x^2 - 5x - 14\).
\(=(x-7)(x+2)\)
3. Factor \(3x^2 + 10x + 8\).
Rewrite: \(3x^2+4x+6x+8\)
Group: \(x(3x+4)+2(3x+4)=(3x+4)(x+2)\)
4. Factor \(5x^2 - 13x - 6\).
Rewrite: \(5x^2+2x-15x-6\)
Group: \(x(5x+2)-3(5x+2)=(5x+2)(x-3)\)
5. Factor completely: \(2x^3 + 16x^2 + 30x\).
Then factor trinomial: \(2x(x+3)(x+5)\)
Factoring by Grouping
When to Use
Grouping works best for 4-term polynomials. It's also used as the last step in the AC Method for trinomials with \(a \ne 1\).
Steps
- Check for a GCF first and factor it out.
- Group the four terms into two pairs.
- Factor the GCF out of each pair separately.
- If both pairs now share a common binomial factor, factor that out.
- If they don't match, try rearranging the terms and grouping differently.
Example 1
Factor \(x^3 + 2x^2 + 3x + 6\).
Group: \((x^3 + 2x^2) + (3x + 6)\)
Factor each: \(x^2(x+2) + 3(x+2)\)
Common factor \((x+2)\): \(= (x+2)(x^2+3)\)
Example 2 — Rearrange needed
Factor \(ax + 2b + 2a + bx\).
Rearrange: \(ax + 2a + bx + 2b\)
Group: \((ax+2a) + (bx+2b)\)
Factor each: \(a(x+2) + b(x+2)\)
Common factor: \(= (x+2)(a+b)\)
Example 3 — With negative
Factor \(x^3 - x^2 - 4x + 4\).
Group: \((x^3 - x^2) + (-4x + 4)\)
Factor each: \(x^2(x-1) - 4(x-1)\)
Common factor: \((x-1)(x^2-4) = (x-1)(x+2)(x-2)\)
Practice Problems
1. Factor \(x^3 + 5x^2 + 2x + 10\).
\(= x^2(x+5)+2(x+5) = (x+5)(x^2+2)\)
2. Factor \(2x^3 - 3x^2 + 4x - 6\).
\(= x^2(2x-3)+2(2x-3) = (2x-3)(x^2+2)\)
3. Factor \(x^3 + 3x^2 - 4x - 12\).
\(= x^2(x+3)-4(x+3) = (x+3)(x^2-4) = (x+3)(x+2)(x-2)\)
4. Factor \(6x^2y - 4xy + 9xy^2 - 6y^2\). (Hint: try grouping terms 1&3, then 2&4, or just 1&2 and 3&4)
\(= 2xy(3x-2)+3y^2(3x-2) = (3x-2)(2xy+3y^2) = y(3x-2)(2x+3y)\)
🎯 Factoring Quiz
8 questions — click your answer to check instantly.
1. Factor \(12x^2 - 8x\).
2. Factor \(x^2 - 36\).
3. Factor \(x^2 + 9x + 20\).
4. Factor \(x^2 - 3x - 18\).
5. Factor \(2x^2 + 5x + 3\) using the AC Method.
6. Factor completely: \(3x^2 - 75\).
7. Factor by grouping: \(x^3 + 4x^2 + 3x + 12\).
8. Factor completely: \(x^4 - 81\).