Factor polynomials up to degree four

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Recognize structure, choose a method, and check by expanding

Factoring rewrites a polynomial as a product of simpler expressions. For example, the expression

x^2+5x+6

can be written as

(x+2)(x+3)

. Multiplying the factors returns the original expression. This lesson reviews useful Grade 11 skills, then applies them to polynomials whose degree is at most four. The degree is the greatest exponent of the variable with a nonzero coefficient.

What you will learn

Identify common factors and familiar patterns in polynomial expressions.
Factor quadratic polynomials and selected polynomials of degree three or four.
Choose a factoring method based on the structure of the expression.
Check a factorization by multiplying its factors.

1. Prerequisite bridge: factors and degree

A polynomial is an expression made from terms such as numbers and powers of a variable, combined by addition or subtraction. In this lesson, the variable is usually

x

. A term is one part of an expression separated by addition or subtraction. In

3x^2-5x+2

, the terms are

3x^2

,

-5x

, and

2

.

A factor is an expression that is multiplied by another expression. Since

3(x+2)=3x+6

, the expression

3x+6

has factors

3

and

x+2

. Factoring reverses multiplication and expansion.

The degree of a nonzero polynomial in one variable is its highest exponent. For instance,

4x^3-x+7

has degree three. The expression

2x^4+3x^2-1

has degree four, even though it has no

x^3

or

x

term.

Before factoring, arrange terms in descending powers when that makes the structure easier to see. Look for a number or variable factor shared by every term. Taking out this common factor is often the best first move.

Factoring and expanding undo one another.
The degree is the largest exponent present, not the number of terms.
Check for a common factor before trying another method.

2. Useful factoring patterns

A common factor can include a number, a variable, or both. For example, each term in

6x^3+9x^2

contains

3x^2

. Removing it leaves

3x^2(2x+3)

. The terms inside the brackets must multiply by the common factor to reproduce the original terms.

A difference of squares is a subtraction of two perfect squares. A perfect square is an expression multiplied by itself, such as

x^2

or

25

. The pattern

a^2-b^2=(a-b)(a+b)

applies when both parts are squares and are separated by subtraction.

For a quadratic trinomial of the form

x^2+bx+c

, seek two numbers whose product is

c

and whose sum is

b

. Those numbers become the constants in two binomial factors. A binomial is a polynomial with two terms.

For expressions with four terms, grouping can help. Group pairs of terms, factor a common factor from each pair, and look for a matching bracket. If the brackets match, that bracket is a common factor.

A polynomial of degree four may also be viewed as a quadratic in

x^2

. If every variable power is even, let

u=x^2

temporarily. Factor the resulting quadratic in

u

, then replace each

u

with

x^2

and continue if a factor pattern applies.

a^2-b^2=(a-b)(a+b)

Take out a greatest common factor first.
Use difference of squares only for a subtraction of two squares.
For $x^2+bx+c$ , match the product and sum.
Grouping is useful when pairs produce the same bracket.
An expression in $x^2$ can sometimes be factored as a quadratic in a temporary variable.

3. A method for choosing and checking

No single method works for every polynomial. Start by counting the terms and checking for a common factor. Then inspect the remaining expression for a familiar pattern.

For a trinomial of degree two, use the product-and-sum method when the coefficient of

x^2

is one. For four terms, test grouping. For a difference of squares, use the matching pattern. For a degree-four expression with only even powers, consider treating it as a quadratic in

x^2

.

After factoring, multiply the factors to check the result. Expand one pair at a time, combine like terms, and compare with the original polynomial. Like terms have the same variable raised to the same power. For example,

2x^2

and

5x^2

are like terms, but

2x^2

and

5x

are not.

A polynomial is fully factored over the integers when it has been written as a product and none of its factors can be factored further using integer coefficients. For this lesson, report the common factor as well as any further factorization that the expression allows.

Choose a method by looking at the expression's structure.
Do not stop after removing a common factor if the remaining factor can be factored.
Expansion is a reliable check for a proposed factorization.

4. Apply the patterns carefully

Factoring can involve more than one decision. A common factor may reveal a familiar pattern that was not obvious at first. Similarly, a quadratic-in-

x^2

approach may produce a difference of squares that needs another step.

Keep the variable and exponents visible while working. Replacing

x^2

by a temporary letter can make a degree-four expression easier to read, but it does not change the original variable. After factoring, substitute back and check whether the new factors can be broken down further.

Signs matter. In a quadratic trinomial, the product determines whether the two constants have matching or opposite signs, and their sum determines which signs are needed. Always verify both conditions rather than relying on a guess.

A first factoring step may expose a second one.
Replace a temporary variable before giving the final factorization.
Check both the product and sum when factoring a monic quadratic trinomial.

Choose a factoring approach

What you notice	Approach to try	Example pattern
Every term shares a factor	Take out the common factor	$6x^2+9x=3x(2x+3)$
Two squares are separated by subtraction	Use difference of squares	$x^2-16=(x-4)(x+4)$
A monic quadratic trinomial	Find a product and sum pair	$x^2+7x+12=(x+3)(x+4)$
Four terms can form matching brackets	Group pairs of terms	$ax+ay+bx+by=(a+b)(x+y)$
Only even powers appear in a degree-four expression	Treat it as a quadratic in $x^2$	$x^4+bx^2+c$

Worked example

Factor a degree-four polynomial

Factor

x^4-5x^2+4

fully over the integers.

Notice the structure
The expression has degree four, but its variable powers are $4$ , $2$ , and $0$ . Treat it as a quadratic in $x^2$ by using $u$ in place of $x^2$ .
$u=x^2$
Factor the quadratic
We need two numbers with product $4$ and sum $-5$ . The numbers $-1$ and $-4$ meet both conditions, so the quadratic factors as shown.
$u^2-5u+4=(u-1)(u-4)$
Substitute back
Replace each $u$ with $x^2$ . Both new factors are differences of squares, so each can be factored again.
$(x^2-1)(x^2-4)$
Factor each difference of squares
Apply the difference-of-squares pattern to both brackets. The four resulting linear factors cannot be factored further over the integers.
$(x-1)(x+1)(x-2)(x+2)$

Answer:

x^4-5x^2+4=(x-1)(x+1)(x-2)(x+2)

Check: Pair the factors as

(x^2-1)(x^2-4)

. Their product is

x^4-5x^2+4

, which matches the original.

Common mistakes and how to avoid them

Stopping after taking out a common factor.

Correction: Check the remaining bracket for another factorization. A complete answer may require several stages.

Using the difference-of-squares pattern on a sum, such as

x^2+9

.

Correction: The pattern requires subtraction between the squares. Do not change a plus sign to a minus sign.

Choosing two numbers whose product is correct but whose sum is not.

Correction: For

x^2+bx+c

, check both conditions: the product must be

c

and the sum must be

b

.

Forgetting to replace a temporary variable such as

u

with the original expression.

Correction: Substitute back before writing the final factors in terms of

x

.

Changing a sign while expanding to check the result.

Correction: Multiply each pair carefully, then combine like terms and compare every coefficient with the original.

Lesson summary

Factoring rewrites a polynomial as a product.
Check for a common factor before using another method.
Use product and sum for monic quadratic trinomials, grouping for suitable four-term expressions, and difference of squares for a subtraction of two squares.
A degree-four expression with only even powers may be treated as a quadratic in $x^2$ .
Expand the final factors to verify the result.

Check your understanding

Question 1

Which is the fully factored form of

x^2-9

over the integers?

$(x-3)(x+3)$
$(x-9)(x+1)$
$(x-3)^2$
$x(x-9)$

Show answer and explanation

(x-3)(x+3)

x^2-9

is a difference of squares:

x^2-3^2=(x-3)(x+3)

. Multiplying the factors gives

x^2-9

.

Question 2

Which pair of numbers factors

x^2+x-12

as a product of two binomials?

$4$ and $-3$
$-4$ and $3$
$6$ and $-2$
$-6$ and $2$

Show answer and explanation

4

and

-3

4(-3)=-12

and

4+(-3)=1

, so

x^2+x-12=(x+4)(x-3)

.

Question 3

What should be done first when factoring

8x^3-12x^2

?

Take out the common factor $4x^2$ .
Use the difference-of-squares pattern.
Treat the expression as a quadratic in $x^2$ .
Group the two terms into pairs.

Show answer and explanation

Take out the common factor

4x^2

.

Both terms share

4x^2

, so the first step is

8x^3-12x^2=4x^2(2x-3)

. The remaining binomial does not factor further over the integers.

Key terms

Polynomial: An expression made from terms with numerical coefficients and nonnegative integer powers of a variable, joined by addition or subtraction.
Term: One part of an expression separated from other parts by addition or subtraction.
Degree: The greatest exponent of the variable in a nonzero polynomial.
Factor: A quantity or expression multiplied by another to make a product.
Binomial: A polynomial with two terms.
Monic quadratic: A quadratic polynomial whose coefficient of $x^2$ is $1$ .
Difference of squares: A subtraction of two perfect-square expressions that can be factored as the difference and sum of their square roots.
Like terms: Terms with the same variable raised to the same power.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C3.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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