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RF11 · Factor polynomials of degree three through five with integer coefficients

Learn to factor polynomials of degree three through five with integer coefficients through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Relations and Functions

Recognize structure, find a factor, and keep checking your work

A polynomial is an expression made from terms with variables raised to whole-number powers. Its degree is the greatest exponent of the variable. For example, 3x4−2x+13x^4-2x+1 has degree four. To factor a polynomial means to write it as a product of simpler polynomial expressions. This lesson focuses on polynomials of degree three, four, or five with integer coefficients. The goal is to choose a method that matches the expression, then verify the result by expanding.

What you will learn

  • Recognize a greatest common factor and factor it out.
  • Factor polynomials by grouping and by using familiar products.
  • Use integer candidates to find a linear factor of a polynomial.
  • Check a factorization by multiplying the factors.

1. Review: factors, products, and common factors

A factor is an expression multiplied by another expression. Since 6x=2(3x)6x=2(3x), both 22 and 3x3x are factors of 6x6x. Factoring reverses expansion: it rewrites a sum as a product.
Before factoring, look for a greatest common factor, or GCF. The GCF is the largest numerical and variable factor shared by every term. For example, the terms in 6x3+9x26x^3+9x^2 share 3x23x^2, so 6x3+9x2=3x2(2x+3)6x^3+9x^2=3x^2(2x+3). Factoring out a GCF is often the best first step.
You should also recognize difference of squares: a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b). A sum of squares, such as x2+4x^2+4, does not factor into two linear factors with integer coefficients.
a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
  • Check for a GCF before trying another method.
  • After factoring, check whether any factor can be factored further using the methods in this lesson.

2. Match the polynomial to a method

Grouping works well when terms can be arranged into pairs that share a common bracket. For instance, x3+3x2−4x−12x^3+3x^2-4x-12 can be grouped as (x3+3x2)+(−4x−12)(x^3+3x^2)+(-4x-12). Factoring each pair gives x2(x+3)−4(x+3)x^2(x+3)-4(x+3). The repeated bracket is then a factor.
A polynomial may also contain a familiar expression. In x4−5x2+4x^4-5x^2+4, treat x2x^2 as one quantity: the expression has the same pattern as a quadratic. It factors as (x2−1)(x2−4)(x^2-1)(x^2-4), and each factor is a difference of squares.
For other polynomials, test integer values that could make the polynomial equal zero. If substituting an integer rr gives zero, then (x−r)(x-r) is a factor. For a polynomial with integer coefficients, possible integer roots must be factors of the constant term. This gives a finite list to test. Once a linear factor is found, divide the polynomial by it to obtain the remaining factor.
A root is a value that makes a polynomial equal zero. The factor and root are linked: a root rr corresponds to the factor (x−r)(x-r). This is useful for finding a factor, but a proposed factorization still needs to be checked by multiplication.
P(r)=0\iff (x-r) is a factor of P(x)
  • Use grouping when pairs produce the same bracket.
  • Look for difference of squares, including expressions involving powers such as x2x^2.
  • To test an integer candidate, substitute it into the original polynomial.

3. A reliable checking routine

Write factors in a clear order, and keep signs attached to their terms. A sign error can change every factor. When you use grouping, the two grouped parts must produce exactly the same bracket before you factor that bracket out.
When you test a candidate root, evaluate the complete polynomial. Do not test only its first few terms. If the result is not zero, that candidate does not give the claimed linear factor.
The strongest final check is to expand your factors and compare the result with the original polynomial. Matching the leading term alone is not enough; every coefficient must match.
  • Check every coefficient by expanding.
  • If a factor remains, decide whether it factors further with integer coefficients.

4. Practice before the check

Try these original practice questions. First look for a GCF or a recognizable pattern. If neither is clear, try grouping or test integer candidates. Finish each question by multiplying your factors back together.
Practice: factor x3−2x2−9x+18x^3-2x^2-9x+18. Factor x4−13x2+36x^4-13x^2+36. Factor x5−3x4−4x+12x^5-3x^4-4x+12.
  • Choose a method based on the structure of the expression.
  • A complete answer is fully factored over the integers and verified.

Worked example

Grouping a cubic

Factor x3+3x2−4x−12x^3+3x^2-4x-12 completely.
Cubic and its integer zeros
Cubic and its integer zerosThe curve shows a window of the full graph. Its zeros are at $-3$, $-2$, and $2$.-4-3-2-10123-21-14-7071421P(x)xy

The curve shows a window of the full graph. Its zeros are at $-3$, $-2$, and $2$.

  1. Group the terms
    Put the first two terms together and the last two together. This arrangement makes it possible to factor each pair.
    (x3+3x2)+(−4x−12)(x^3+3x^2)+(-4x-12)
  2. Factor each pair
    The first pair has a common factor of x2x^2. The second pair has a common factor of −4-4. Both resulting terms contain the bracket (x+3)(x+3).
    x2(x+3)−4(x+3)x^2(x+3)-4(x+3)
  3. Factor the repeated bracket
    Since (x+3)(x+3) is a factor of both terms, factor it out. The remaining expression is a difference of squares.
    (x+3)(x2−4)=(x+3)(x−2)(x+2)(x+3)(x^2-4)=(x+3)(x-2)(x+2)
Answer: (x+3)(x−2)(x+2)(x+3)(x-2)(x+2)
Check: The two linear factors involving xx first give x2−4x^2-4. Multiplying by (x+3)(x+3) gives x3+3x2−4x−12x^3+3x^2-4x-12, as required.

Worked example

A quartic with a difference-of-squares pattern

Factor x4−5x2+4x^4-5x^2+4 completely over the integers.
Quartic and its integer zeros
Quartic and its integer zerosThe curve shows a window of the full graph. Its zeros are at $-2$, $-1$, $1$, and $2$.-3-2-10123-8-4048121620P(x)xy

The curve shows a window of the full graph. Its zeros are at $-2$, $-1$, $1$, and $2$.

  1. Spot the repeated power
    The powers of xx are four, two, and zero. Treat x2x^2 as one quantity to factor the expression as a quadratic pattern.
    (x2−1)(x2−4)(x^2-1)(x^2-4)
  2. Factor each difference of squares
    Each bracket is a difference of squares. Apply the difference-of-squares rule to both brackets.
    (x−1)(x+1)(x−2)(x+2)(x-1)(x+1)(x-2)(x+2)
Answer: (x−1)(x+1)(x−2)(x+2)(x-1)(x+1)(x-2)(x+2)
Check: The first pair multiplies to x2−1x^2-1 and the second to x2−4x^2-4. Their product is x4−5x2+4x^4-5x^2+4.

Worked example

Testing integer candidates in a quintic

Factor x5−2x4−x+2x^5-2x^4-x+2 completely over the integers.
Quintic and its integer zeros
Quintic and its integer zerosThe curve shows a window of the full graph. Its integer zeros are at $-1$, $1$, and $2$.-2-10123-24-16-8081624P(x)xy

The curve shows a window of the full graph. Its integer zeros are at $-1$, $1$, and $2$.

  1. List integer candidates
    Any integer root must be a factor of the constant term, 22. Test the integer candidates 11, −1-1, 22, and −2-2. Substituting 22 gives zero, so (x−2)(x-2) is a factor.
    P(2)=32−32−2+2=0P(2)=32-32-2+2=0
  2. Divide by the linear factor
    Dividing by (x−2)(x-2) leaves x4−1x^4-1. This can be factored using the difference-of-squares rule.
    x5−2x4−x+2=(x−2)(x4−1)x^5-2x^4-x+2=(x-2)(x^4-1)
  3. Finish factoring
    Factor x4−1x^4-1 as a difference of squares, then factor x2−1x^2-1 once more. The factor x2+1x^2+1 has no integer linear factors.
    x5−2x4−x+2=(x−2)(x−1)(x+1)(x2+1)x^5-2x^4-x+2=(x-2)(x-1)(x+1)(x^2+1)
Answer: (x−2)(x−1)(x+1)(x2+1)(x-2)(x-1)(x+1)(x^2+1)
Check: The product (x−2)(x2−1)(x-2)(x^2-1) is x3−2x2−x+2x^3-2x^2-x+2. Multiplying by x2+1x^2+1 gives x5−2x4−x+2x^5-2x^4-x+2.

Common mistakes and how to avoid them

Stopping after finding one factor, even when the remaining factor can be factored further.
Correction: Continue factoring each remaining factor over the integers.
Claiming (x−r)(x-r) is a factor without checking that substitution of rr gives zero.
Correction: Evaluate the full polynomial at rr before using the factor.
Losing a negative sign when grouping terms.
Correction: Keep the negative sign with the terms in the second group and check that both groups produce the same bracket.

Lesson summary

  • Start by checking for a GCF.
  • Use grouping when pairs produce a shared bracket.
  • Look for difference of squares, including expressions built from powers of xx.
  • For integer candidates, a zero value identifies a corresponding linear factor.
  • Expand the final factors to verify every term.

Check your understanding

Question 1

Which is the complete factorization of x3−2x2−9x+18x^3-2x^2-9x+18?
  1. (x−2)(x−3)(x+3)(x-2)(x-3)(x+3)
  2. (x+2)(x−3)(x+3)(x+2)(x-3)(x+3)
  3. (x−2)(x2−9)(x-2)(x^2-9)
  4. correctIndex
Show answer and explanation
(x−2)(x−3)(x+3)(x-2)(x-3)(x+3)
Grouping gives x2(x−2)−9(x−2)=(x−2)(x2−9)x^2(x-2)-9(x-2)=(x-2)(x^2-9). The difference of squares then gives (x−2)(x−3)(x+3)(x-2)(x-3)(x+3).

Question 2

Which value is an integer root of x4−13x2+36x^4-13x^2+36?
  1. 3
  2. 5
  3. -6
  4. correctIndex
Show answer and explanation
3
Substitution of 33 gives 81−117+36=081-117+36=0. The expression also factors as (x2−9)(x2−4)(x^2-9)(x^2-4).

Question 3

What is the complete integer factorization of x5−3x4−4x+12x^5-3x^4-4x+12?
  1. (x−3)(x2−4)(x2+1)(x-3)(x^2-4)(x^2+1)
  2. (x−3)(x4−4)(x-3)(x^4-4)
  3. (x+3)(x2−4)(x2+1)(x+3)(x^2-4)(x^2+1)
  4. correctIndex
Show answer and explanation
(x−3)(x2−4)(x2+1)(x-3)(x^2-4)(x^2+1)
Grouping gives x4(x−3)−4(x−3)=(x−3)(x4−4)x^4(x-3)-4(x-3)=(x-3)(x^4-4). Then x4−4=(x2−2)(x2+2)x^4-4=(x^2-2)(x^2+2), so the listed option is not correct. The correct factorization is (x−3)(x2−2)(x2+2)(x-3)(x^2-2)(x^2+2).

Key terms

Polynomial
An expression made from terms with variables raised to whole-number powers.
Degree
The greatest exponent of the variable in a polynomial.
Factor
An expression multiplied by another expression to make a product.
Greatest common factor
The largest factor shared by every term in an expression.
Root
A value that makes a polynomial equal zero.

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