Factor simple trinomials and common-factor quadratics

Ontario Grade 11 Mathematics

Mathematical Models

MBF3C study topic A1.7

Factoring rewrites an expression as a product. It is the reverse of expanding brackets. For example, expanding

3(x+2)

gives

3x+6

; factoring

3x+6

returns

3(x+2)

. In this lesson, you will practise two kinds of factoring: taking out a common factor and factoring simple trinomials. These skills help you rewrite quadratic expressions in a useful form. The focus is on expressions, not on solving quadratic equations.

What you will learn

Recognize when a quadratic expression has a common factor.
Factor a quadratic by first taking out its greatest common factor.
Factor simple trinomials of the form $x^2+bx+c$ by finding a pair of numbers with the correct sum and product.
Check a factorization by expanding the factors.

1. Review: factors, products, and quadratics

A factor is a number or expression that is multiplied by another number or expression. In

4\cdot 5=20

, the numbers

4

and

5

are factors of the product

20

. A bracket can also be a factor:

2(x+3)

means

2

multiplied by the entire expression in the bracket.

A quadratic expression has a variable term with an exponent of

2

, along with any other terms. For example,

x^2+5x+6

is quadratic. In this lesson, the variable is usually

x

. The exponent tells how many times

x

is multiplied by itself, so

x^2=x\cdot x

.

To expand a product of two brackets, multiply each term in one bracket by each term in the other. For instance,

(x+2)(x+3)

expands to

x^2+5x+6

. Factoring reverses this process: it looks for brackets whose product gives the original expression.

Before you factor, arrange the terms in descending powers of the variable. For a quadratic, this usually means writing the squared term first, then the term with

x

, then the constant term. This makes it easier to see the structure.

Factoring and expanding undo each other.
A quadratic expression includes a squared variable term.
Always check a factorization by expanding.

2. Factor out a common factor

A common factor is a factor shared by every term in an expression. The greatest common factor, or GCF, is the largest factor shared by all the terms. To factor by the GCF, identify what divides every term, then write that factor outside a bracket.

For example, both terms in

6x^2+9x

are divisible by

3x

. Dividing each term by

3x

leaves

2x

and

3

. Therefore, the expression becomes

3x(2x+3)

. The bracket contains what remains from each original term.

This method also works when the expression is quadratic. A common-factor quadratic often has two terms, such as

8x^2-12x

, or three terms, such as

4x^2+12x+8

. Take out the GCF first whenever every term shares one. The expression left inside the bracket may itself be a trinomial that can be factored further.

The sign matters. If all terms are negative, factoring out a negative common factor can make the bracket easier to read. For example,

-5x^2-10x=-5x(x+2)

. Expanding confirms that both terms have the original negative signs.

The common factor must divide every term.
Divide each term by the factor you take out to find the bracket.
If possible, factor again after taking out the GCF.

3. Factor simple trinomials

A simple trinomial in this lesson has the form

x^2+bx+c

. A trinomial is an expression with three terms. The coefficient of

x^2

is

1

, so the leading term is just

x^2

. The numbers

b

and

c

can be positive or negative.

To factor

x^2+bx+c

, look for two integers whose product is

c

and whose sum is

b

. The product gives the constant term, while the sum gives the coefficient of

x

. If the two numbers are

m

and

n

, then the factors are

(x+m)(x+n)

.

Why does the sum matter? Expanding

(x+m)(x+n)

gives

x^2+(m+n)x+mn

. The middle coefficient is the sum of the two numbers, and the constant is their product. So checking both conditions makes sure the brackets expand to the original trinomial.

Signs help narrow the search. If the constant is positive, the two numbers have the same sign. Their sum tells you whether both are positive or both are negative. If the constant is negative, the numbers have opposite signs. The larger absolute value determines the sign of their sum.

If no integer pair has both the required product and sum, do not force a factorization using an incorrect pair. For this method, the simple trinomial factors using integer brackets only when a suitable pair exists. Always expand your proposed factors to verify them.

(x+m)(x+n)=x^2+(m+n)x+mn

For $x^2+bx+c$ , find integers with product $c$ and sum $b$ .
The factors are $(x+m)(x+n)$ when $m+n=b$ and $mn=c$ .
A correct product alone is not enough; the sum must also match.

4. Choose a method and check your result

When you see a quadratic expression, first check whether every term has a common factor. If it does, take out the GCF. Then look at the expression inside the bracket to see whether another factor can be taken out or whether the remaining expression is a simple trinomial.

If there is no common factor and the expression is a simple trinomial with leading coefficient

1

, use the sum-and-product search. List factor pairs of the constant and test their sums. This is more reliable than guessing based only on the middle term.

After factoring, expand the result. Multiply the first terms, the outer terms, the inner terms, and the last terms. Combine like terms and compare with the starting expression. Like terms have the same variable part, such as

2x

and

5x

; they can be combined by adding or subtracting their coefficients.

Factoring changes the form of an expression but does not change its value. The factored and expanded forms represent the same expression. This is why expansion is a useful check, and why a missing sign or factor will show up when you compare the result.

Look for a GCF before using the trinomial method.
Factor fully when the remaining expression can be factored further.
Expand the final factors to confirm they reproduce the original terms.

Worked example

Example 1: Common-factor quadratic

Factor

12x^2-18x

fully.

Find the shared factor
The coefficients $12$ and $18$ share a greatest numerical factor of $6$ . Each term also contains at least one $x$ , so the GCF is $6x$ .
$6x$
Divide each term
Divide each original term by $6x$ . The first term leaves $2x$ , and the second leaves $-3$ . Put these results inside the bracket.
$12x^2-18x=6x(2x-3)$
Check by expanding
Multiply $6x$ by both terms in the bracket. The result matches both terms of the original expression, including the negative sign.
$6x(2x-3)=12x^2-18x$

Answer:

6x(2x-3)

Check: Expanding gives

12x^2-18x

, so the factorization is correct.

Worked example

Example 2: Simple trinomial

Factor

x^2-x-20

.

Identify the target sum and product
The coefficient of $x$ is $-1$ , and the constant is $-20$ . We need two integers that multiply to $-20$ and add to $-1$ .
$mn=-20,\qquad m+n=-1$
Choose a pair
The pair $-5$ and $4$ has product $-20$ and sum $-1$ . Since the product is negative, the pair has opposite signs, which fits the negative middle coefficient.
$(-5)(4)=-20,\qquad -5+4=-1$
Write the factors
Place each selected integer with $x$ in a bracket. The brackets multiply to the original trinomial because their cross-products combine to give the coefficient $-1$ .
$(x-5)(x+4)$
Check by expanding
The cross-products are $4x$ and $-5x$ , which combine to $-x$ . The constant is $-20$ , so all three terms match.
$(x-5)(x+4)=x^2-x-20$

Answer:

(x-5)(x+4)

Check: Expanding gives

x^2-x-20

, matching the original expression.

Common mistakes and how to avoid them

Taking out a factor that is not common to every term.

Correction: Check each term separately. The factor must divide all terms exactly.

Choosing numbers whose product is correct but whose sum is wrong.

Correction: For a simple trinomial, verify both the product and the sum before writing brackets.

Losing a negative sign when dividing a term or writing a bracket.

Correction: Keep the signs attached to their terms, then expand the proposed factors to check.

Stopping after taking out a common factor when the bracket can be factored further.

Correction: Look at the expression inside the bracket and check whether it has another common factor or is a factorable simple trinomial.

Lesson summary

Factoring rewrites an expression as a product and can be checked by expanding.
For a common-factor quadratic, take out the greatest factor shared by every term.
For $x^2+bx+c$ , find two integers with product $c$ and sum $b$ .
Check coefficients, constants, and signs in the expanded result.

Check your understanding

Question 1

Factor

10x^2+15x

fully.

$5x(2x+3)$
$5(2x^2+3x)$
$10x(x+15)$
$5x(2x-3)$

Show answer and explanation

5x(2x+3)

The GCF is

5x

. Dividing each term by

5x

leaves

2x

and

3

, so

5x(2x+3)

expands to the original expression.

Question 2

Which is the factorization of

x^2+2x-15

?

$(x+5)(x-3)$
$(x-5)(x+3)$
$(x+15)(x-1)$
$(x+3)(x-5)$

Show answer and explanation

(x+5)(x-3)

The numbers

5

and

-3

multiply to

-15

and add to

2

. Therefore,

(x+5)(x-3)

is correct.

Question 3

Which pair can be used to factor

x^2-7x+12

?

$-3$ and $-4$
$3$ and $4$
$-2$ and $-6$
$-1$ and $-12$

Show answer and explanation

-3

and

-4

The pair

-3

and

-4

has product

12

and sum

-7

, giving

(x-3)(x-4)

.

Key terms

Factor: A number or expression multiplied by another to make a product.
Quadratic expression: An expression that includes a variable term with exponent $2$ .
Trinomial: An algebraic expression with three terms.
Greatest common factor: The largest factor shared by every term in an expression.
Expand: Multiply factors and brackets to rewrite a product as a sum or difference of terms.

Continue through MBF3C

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.7. 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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