Factoring Completely

Learn to combine all factoring techniques to factor any polynomial expression completely.

Advanced25 minLesson
Printable worksheet

Definition

Factoring completely means breaking down a polynomial into the product of prime factors (factors that cannot be factored further).
To factor completely, follow this systematic approach:
Step 1: Always look for a GCF first
2x3+6x2=2x2(x+3)2x^3 + 6x^2 = 2x^2(x + 3)
Step 2: Count the terms and apply appropriate technique
  • 2 terms: Check for difference of squares, sum/difference of cubes
  • 3 terms: Try trinomial factoring methods
  • 4+ terms: Try factoring by grouping
Step 3: Check if any factor can be factored further
x4−16=(x2+4)(x2−4)=(x2+4)(x+2)(x−2)x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)
A polynomial is completely factored when no factor can be factored any further using integer coefficients.

Try it now

What is the first step when factoring any polynomial completely?

Worked Examples

Factor completely: 3x3−12x3x^3 - 12x
1

Find the GCF

GCF of 3x33x^3 and 12x12x is 3x3x → 3x(x2−4)3x(x^2 - 4)
2

Check remaining factor

x2−4x^2 - 4 is a difference of squares: a2−b2a^2 - b^2 where a=xa = x, b=2b = 2 → Can be factored further
3

Factor the difference of squares

x2−4=(x+2)(x−2)x^2 - 4 = (x + 2)(x - 2)
4

Write complete factorization

Combine GCF with factored form → 3x(x+2)(x−2)3x(x + 2)(x - 2)

Common Mistakes

Forgetting to factor out the GCF first

Why it's wrong: Starting with other techniques when there's a GCF makes the problem harder and can lead to incomplete factoring.

Correct: Always check for GCF before any other factoring technique. 6x2−24=6(x2−4)=6(x+2)(x−2)6x^2 - 24 = 6(x^2 - 4) = 6(x+2)(x-2)

Stopping too early

Why it's wrong: After one factoring step, students often forget to check if factors can be factored further.

Correct: Always examine each factor. x4−1=(x2+1)(x2−1)x^4 - 1 = (x^2+1)(x^2-1) but (x2−1)=(x+1)(x−1)(x^2-1) = (x+1)(x-1), so the complete answer is (x2+1)(x+1)(x−1)(x^2+1)(x+1)(x-1)

Trying to factor a sum of squares

Why it's wrong: a2+b2a^2 + b^2 cannot be factored using real numbers (no real factors).

Correct: Only difference of squares factors: x2−9=(x+3)(x−3)x^2 - 9 = (x+3)(x-3), but x2+9x^2 + 9 is already prime.

Incorrect sign in grouping

Why it's wrong: When factoring a negative from a group, students forget to change signs inside.

Correct: −2x−6=−2(x+3)-2x - 6 = -2(x + 3), not −2(x−3)-2(x - 3). Check by distributing back!

Interactive Visual

Factor Tree

Number:
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Enter a number to see its factor tree and prime factorization.

Function graph

-3-2-10123-20-1001020xy(-2, 0)(0, 0)(2, 0)

Tap or hover over the graph to read off values.

3x^3 - 12x3x(x + 2)(x - 2)
  • Zero (x-intercept): (-2, 0)
  • Zero (x-intercept): (0, 0)
  • Zero (x-intercept): (2, 0)
  • Zero (x-intercept): (-2, 0)
  • Zero (x-intercept): (0, 0)
  • Zero (x-intercept): (2, 0)

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the first step when factoring any polynomial completely?

Why It Matters

Factoring completely is essential because:
  • Solving equations: To find all solutions to polynomial equations, you need complete factorization
  • Simplifying expressions: Reducing fractions requires factoring both numerator and denominator completely
  • Real-world modeling: Physics, engineering, and economics use factored forms to find critical points
  • Foundation for calculus: Finding zeros of functions requires complete factorization
Mastering this skill combines everything you've learned about factoring into one powerful problem-solving strategy!

Real World Applications

Engineering: Projectile Motion

Engineers use complete factorization to find when and where objects land.

Example:

A ball's height is modeled by h=−16t2+48th = -16t^2 + 48t. Factor to find when it hits the ground: −16t(t−3)=0-16t(t - 3) = 0, so t=0t = 0 or t=3t = 3 seconds.

1Try It Yourself

A rocket's height is h=−4t2+16t+20h = -4t^2 + 16t + 20. To find when it lands, we set h=0h = 0.

Factor −4t2+16t+20-4t^2 + 16t + 20 completely.

Step 1: Write the mathematical expression

First factor out the GCF:

Computer Science: Algorithm Optimization

Programmers factor expressions to simplify calculations and improve efficiency.

Example:

Instead of computing n3−nn^3 - n separately, factor to n(n−1)(n+1)n(n-1)(n+1) - the product of three consecutive integers!

2Try It Yourself

A sorting algorithm's comparison count is C=n2−1C = n^2 - 1.

Factor this expression completely.

Step 1: Write the mathematical expression

Recognize the pattern:

Key Takeaways

  • 1Always start by factoring out the Greatest Common Factor (GCF)
  • 2After removing the GCF, identify the number of terms: 2 terms (special patterns), 3 terms (trinomial methods), 4+ terms (grouping)
  • 3Check each factor to see if it can be factored further
  • 4Sum of squares (a2+b2a^2 + b^2) cannot be factored with real numbers
  • 5A polynomial is completely factored when all factors are prime

Frequently Asked Questions

A polynomial is completely factored when: (1) the GCF is factored out, (2) no factor is a difference of squares that can be factored, (3) no trinomial factor can be factored further, and (4) all remaining factors are prime (linear or irreducible quadratic).
A polynomial is completely factored when: (1) the GCF is factored out, (2) no factor is a difference of squares that can be factored, (3) no trinomial factor can be factored further, and (4) all remaining factors are prime (linear or irreducible quadratic).
Regular factoring might stop after one step. Factoring completely means continuing until no factor can be broken down further. For example, x4−16=(x2+4)(x2−4)x^4 - 16 = (x^2 + 4)(x^2 - 4) is factored, but (x2+4)(x+2)(x−2)(x^2 + 4)(x + 2)(x - 2) is factored completely.
The sum of two squares, a2+b2a^2 + b^2, has no real factors. Unlike a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b), there's no similar formula for sums. The expression x2+4x^2 + 4 is called 'prime' or 'irreducible' over the real numbers.

Glossary

Completely factored
A polynomial written as a product of prime factors that cannot be factored further using integer coefficients
Prime polynomial
A polynomial that cannot be factored into polynomials of lower degree with integer coefficients
GCF (Greatest Common Factor)
The largest expression that divides evenly into all terms of a polynomial
Irreducible
A polynomial that cannot be factored over a given number system (e.g., x2+1x^2 + 1 is irreducible over real numbers)

Formula Card

Factoring Strategy Order

1.GCF→2.Count  terms→3.Apply  technique→4.Check  all  factors1. GCF \rightarrow 2. Count\;terms \rightarrow 3. Apply\;technique \rightarrow 4. Check\;all\;factors

The systematic approach to factor any polynomial completely

Difference of Squares

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

Use when you have two perfect squares separated by subtraction

Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Use when you have two perfect cubes added together

Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Use when you have two perfect cubes separated by subtraction

Perfect Square Trinomials

a2±2ab+b2=(a±b)2a^2 \pm 2ab + b^2 = (a \pm b)^2

Trinomials that are the square of a binomial

Today's challengeOne problem, three tries

More in This Topic

Introduction to Factoring

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Factoring Out the GCF

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Factoring by Grouping

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Factoring Trinomials (a=1)

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