Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Factoring with Common Factors and by Grouping

3.2About 65 minutes

Learning goals

I can find the greatest common factor of a polynomial, including variables.
I can factor out a monomial or binomial common factor.
I can factor a four-term expression by grouping.
I can check a factorization by expanding.

Words to know

factor (verb)
Write an expression as a product. Factoring undoes expanding.

greatest common factor (GCF)
The largest expression that divides evenly into every term of a polynomial.

factored fully
Factored so that no factor can be factored any further.

binomial common factor
A two-term bracket, such as (x − 4), that appears in every term and can be factored out.

factoring by grouping
Factoring a four-term polynomial by common factoring pairs of terms, then factoring out the matching bracket.

Running the movie backwards

Imagine watching a video of someone building a snow fort, but played in reverse: the fort comes apart into the exact snow blocks it was built from. Factoring is expanding played in reverse.

Expanding: 2x(x + 2) → 2x² + 4x (a product becomes a sum).
Factoring: 2x² + 4x → 2x(x + 2) (a sum becomes a product).

Why bother? Because a product has a superpower: if a product equals zero, one of its factors must be zero. That is how you will solve quadratic equations later in this unit. Factoring is the doorway.

Remember from Grade 9

In Grade 9 you factored expressions like 12x + 18 by finding the greatest common factor (GCF): 12x + 18 = 6(2x + 3). You also learned to check a factorization by expanding it. Grade 10 extends this to variables in the GCF, to binomial common factors, and to factoring by grouping.

Step 1 of every factoring problem: look for a common factor

The greatest common factor (GCF) of the terms is the largest expression that divides into every term.

Find the GCF of the coefficients (the numbers).
For each variable that appears in every term, take the lowest power.
Multiply these together to get the GCF.
Divide each term by the GCF and write the results inside brackets.
Check by expanding.

For 6x²y and 9xy²: the GCF of 6 and 9 is 3; the lowest power of x is x¹; the lowest power of y is y¹. So the GCF is 3xy.

Factoring as a missing side problem. The area 2x² + 4x is known, and so is one side, 2x (the GCF). Each box divided by 2x gives the other side: x + 2.

Worked example

Example 1: A monomial common factor

Factor 2x² + 4x fully.

GCF of 2 and 4 is 2. Both terms contain x; lowest power is x¹. GCF = 2x.
Divide: 2x² ÷ 2x = x; 4x ÷ 2x = 2.
Write 2x(x + 2).
Check: 2x(x + 2) = 2x² + 4x ✓.

2x(x + 2)

Worked example

Example 2: Three terms and two variables

Factor 12x³y − 18x²y² + 6xy fully.

GCF of 12, 18 and 6 is 6. Lowest power of x is x¹; lowest power of y is y¹. GCF = 6xy.
Divide each term: 12x³y ÷ 6xy = 2x²; −18x²y² ÷ 6xy = −3xy; 6xy ÷ 6xy = 1.
Do not forget the 1: the last term does not disappear.
Check: 6xy(2x² − 3xy + 1) = 12x³y − 18x²y² + 6xy ✓.

6xy(2x² − 3xy + 1)

Worked example

Example 3: A negative common factor

Factor −4x² + 8x with a negative GCF.

When the first term is negative, it is common to take out a negative GCF.
GCF = −4x. Divide: −4x² ÷ (−4x) = x; 8x ÷ (−4x) = −2.
Check: −4x(x − 2) = −4x² + 8x ✓.

−4x(x − 2). (4x(2 − x) is also correct.)

When the common factor is a binomial

A common factor does not have to be a single term. In 3x(x − 4) + 5(x − 4), both terms contain the bracket (x − 4). Treat the whole bracket like one block and factor it out:

3x(x − 4) + 5(x − 4) = (x − 4)(3x + 5).

It is just like 3A + 5A = A(3 + 5), with A standing for (x − 4).

Factoring by grouping

Some four-term expressions have no factor common to all four terms, but pairs of terms do. Then you can group them.

Split the four terms into two pairs.
Common factor each pair.
If the brackets match, factor out the bracket.
If they do not match, try another pairing, or take out a negative from the second pair.

Factoring by grouping in three steps. The brackets in the middle step must match before you can finish.

Worked example

Example 4: Factor by grouping

Factor 2x − 2y + ax − ay.

Group: (2x − 2y) + (ax − ay).
Factor each pair: 2(x − y) + a(x − y).
The brackets match, so factor out (x − y): (x − y)(2 + a).
Check: (x − y)(2 + a) = 2x + ax − 2y − ay ✓ (same terms, different order).

(x − y)(2 + a)

Worked example

Example 5: Grouping a quadratic (a preview)

Factor x² + 3x − 2x − 6 by grouping.

Group: (x² + 3x) + (−2x − 6).
Factor each pair. Take out −2 from the second pair so the bracket matches: x(x + 3) − 2(x + 3).
Factor out (x + 3): (x + 3)(x − 2).
Check: (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6 ✓.

(x + 3)(x − 2). Notice that x² + 3x − 2x − 6 is just x² + x − 6 with the middle term split in two. You will use this idea in the next two lessons.

Factoring checklist (so far)

Always look for a common factor first.
Take out the greatest common factor, so nothing more can be factored from the bracket.
With four terms, try grouping.
Always check by expanding.

Hands-on activity

Build the rectangle

You need: paper, a pencil and a ruler. A grown-up can help if you choose to cut paper with scissors; tearing along a folded edge also works.

Make 2 large squares (about 6 cm by 6 cm). Each one is an x²-tile.
Make 6 long strips (6 cm by 1 cm). Each one is an x-tile (as long as the side of the square).
Arrange all the pieces into one rectangle. Record its length and width.
Find a second rectangle using all the pieces.
Which rectangle matches the GCF factoring of 2x² + 6x?

Model answer: one rectangle is 2x by (x + 3) (stack the two squares with 3 strips beside each). Another is x by (2x + 6) (the squares in a row with all 6 strips beside them). Both are correct products, but only 2x(x + 3) is factored fully: in x(2x + 6), the bracket still has a common factor of 2.

Just for fun

A teacher asked, "What is the greatest common factor of a pizza and a calzone?"

The class agreed: dough. Then they factored it out of lunch.

Your turn

Factor 5x² − 15x fully.

Factor 8a²b + 12ab² fully.

Factor 6x³ − 9x² + 3x fully.

Factor −10x² − 25x using a negative GCF.

Factor x(x + 7) − 4(x + 7).

Factor 3x + 3y + kx + ky by grouping.

Factor x² − 5x + 2x − 10 by grouping.

More practice: warm-up

Factor 4x + 20.

Factor 7x² + 14x.

What is the GCF of 12x² and 18x?

Factor x² − 9x.

Factor 2(a + 3) + b(a + 3).

More practice: on your own

Factor 15m²n³ − 20m³n² fully.

Factor ax − ay − bx + by.

Factor 2x² + 6x + 5x + 15.

Factor 4x(2x − 1) − (2x − 1).

A rectangle has area 3x² + 12x and width 3x. Find its length and a simplified expression for its perimeter.

Challenge

Factor x³ + 2x² + 3x + 6.

Factor 6xy − 9x + 4y − 6.

The surface area of a closed cylinder is 2πr² + 2πrh. Factor this formula, then use it to find the surface area when r = 3 cm and h = 7 cm (to one decimal place).

Check your understanding

Quiz

What is the GCF of 9x²y and 12xy²?

Which shows 6x² − 12x factored fully?

Factor y(x + 4) + 3(x + 4).

Factor 2x − 2y + ax − ay by grouping.

Which expression is NOT equivalent to −4x² + 8x?

What is the best way to check a factorization?

For parents and teachers

Big idea: Expectation 3.2 asks students to factor polynomial expressions involving common factors, trinomials and differences of squares, including grouping examples like 2x − 2y + ax − ay. This lesson covers common factors and grouping; the next three lessons cover trinomials and special products.

How to help: Ask "What do all the terms have in common?" before your teen starts. Then ask them to expand their answer to check. Doing the paper-tile activity together makes the idea of a missing side length very concrete.

Common difficulties:

Not taking out the greatest common factor (e.g. 2(3x² − 6x) instead of 6x(x − 2)).
Dropping a term that becomes 1: 6xy(2x² − 3xy) is missing its + 1.
Sign errors in grouping: the second pair often needs a negative common factor so the brackets match.