Integrated Math 2–3 Honors · Lesson Review · September 22, 2026

Factoring Polynomials with GCF and Grouping

Factoring reverses multiplication: first remove the greatest common factor, then look for a common binomial created by grouping.

39-minute edited lesson10 chaptersCaptions + transcript

01 · Watch

Lesson video

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0:0001Factoring Polynomials with GCF and Grouping 0:0502Multiplication and Its Reverse 1:1003Factor vs. To Factor 3:3604Prime Factorization 13:3705Greatest Common Factors 18:5106Factor Out the GCF 24:3607Factoring by Grouping 31:0808Factor Completely: GCF Then Group 34:1509A Negative Factor in a Group 37:4810Final Four-Term Practice

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02 · Understand

Learning targets

Distinguish a factor from the action to factor.
Use prime factorization to find numerical and monomial GCFs.
Factor a greatest common factor from every term of a polynomial.
Factor four-term polynomials by grouping and factor completely.

03 · Connect

Key ideas

Every expression is rendered with KaTeX plus semantic MathML for clear reading and assistive technology.

01A factor: $24=1\cdot24=2\cdot12=3\cdot8=4\cdot6$ A factor is a number or expression that divides another quantity evenly. The pieces in a product are also called factors.
02To factor: $3x^2(x^2-4x+6)=3x^4-12x^3+18x^2$ To factor means to rewrite an expression as a product. It reverses distribution or polynomial multiplication.
03Prime factorization: $72=2^3\cdot3^2$ Break each number into prime factors. Exponents record repeated factors.
04Greatest common factor: $\operatorname{GCF}(72x^3y^2,18xy^4z^3)=18xy^2$ Use every factor common to all terms. For a shared variable, choose the smallest exponent.
05Grouping: $3x^3-4x^2+6x-8=x^2(3x-4)+2(3x-4)=(x^2+2)(3x-4)$ Factor each pair until the same binomial appears, then factor that common binomial.

04 · Apply

Worked examples

These examples follow the numerical, monomial, GCF, and grouping work developed on the board.

GCF

Start with common factors

20=2^2\cdot5,\quad24=2^3\cdot3\quad\Longrightarrow\quad\operatorname{GCF}=2^2=4

List factors or compare prime factorizations. The greatest number that divides both values is four.

Monomials

Combine the numerical and variable GCFs

\operatorname{GCF}(72x^3y^2,18xy^4z^3)=18xy^2

The numerical GCF is eighteen; the smallest shared powers are $x$ and $y^2$ .

Factor the GCF

Divide every term by the GCF

3x^4-12x^3+18x^2=3x^2(x^2-4x+6)

Check the result by distributing $3x^2$ back across the parentheses.

Variable GCF

Use the smallest shared exponent

-4x^3y+8x^2y^4-18x^4y^2=2x^2y(-2x+4y^3-9x^2y)

The GCF itself is positive. Later, when grouping, a negative factor can make two binomial factors match: $-18x-48=-6(3x+8)$ .

Grouping

Create one common binomial

3x^3-4x^2+6x-8=x^2(3x-4)+2(3x-4)=(x^2+2)(3x-4)

After factoring each pair, $3x-4$ is the shared factor.

Factor completely

GCF first, grouping second

90x^3-120x^2+120x-160=10(3x^2+4)(3x-4)

Continue until no factor can be broken down further using the lesson’s methods.

More practice

Group without a numerical GCF

15x^3+40x^2-18x-48=(5x^2-6)(3x+8)

45x^4+60x^3-15x^2-20x=5x(3x^2-1)(3x+4)

Choose groups that expose the same binomial factor, then check by multiplying.

05 · Avoid

Common mistakes

Stopping after finding the GCF.

Factoring asks for a product, so write the GCF outside parentheses and the quotients inside.

Using the largest variable exponent.

The GCF uses the smallest exponent shared by every term.

Losing a sign.

Divide each complete term, including its sign, by the chosen factor.

Skipping the GCF before grouping.

Always look for a common factor across all terms first.

Getting opposite binomials.

Factor a negative from one group when needed so the binomial factors match exactly.

Failing to check.

Multiply the factors back together to confirm the original expression.

06 · Check yourself

Quick check

Try each prompt before opening the worked answer.

1What is the difference between “a factor” and “to factor”?

A factor is one piece of a product. To factor is to rewrite an expression as a product.

2Why does the variable part of a GCF use the smaller exponent?

That is the greatest power guaranteed to divide every term. See $\operatorname{GCF}(72x^3y^2,18xy^4z^3)=18xy^2$ .

3What should you do before factoring by grouping?

First factor any GCF shared by all terms. Then split the remaining four terms into two pairs.

4How can you verify a factorization?

Multiply the factors. The expanded result must equal the original expression, as illustrated by $3x^2(x^2-4x+6)=3x^4-12x^3+18x^2$ .

07 · Revisit

Transcript and captions

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