Factoring Expressions by Greatest Common Factor

Hello again. In the previous lesson, you solved linear equations by keeping both sides balanced and checking an answer by substitution. This lesson uses some of the same skills—especially division and the distributive property—but it has a different goal.

Here, you will factor — לפרק לגורמים — an algebraic expression — ביטוי אלגברי. There is usually no equal sign and no value of $x$ to find. Instead, you will rewrite an expression as multiplication by taking out its greatest common factor (GCF) — גורם משותף גדול ביותר.

By the end, you should be able to find the GCF of numbers and variables, place it outside parentheses, divide every term correctly, and check your factoring by distributing.

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Factoring is the reverse of distributing

Last lesson, you used the distributive property to expand parentheses:

$$3(2x+5)=6x+15$$

Factoring reverses that process:

$$6x+15=3(2x+5)$$

Both forms have exactly the same value. The first is expanded; the second is in factored form — צורה מפורקת לגורמים.

The number $3$ is a factor — גורם — of both terms because:

$$6x÷3=2x$$

and

$$15÷3=5$$

So $3$ can be taken outside the parentheses.

Important vocabulary for test questions:

English	Hebrew	Meaning
expression	ביטוי	Math with no equal sign, such as $6x+15$
term	איבר	A part separated by $+$ or $-$, such as $6x$ and $15$
factor	גורם	Something being multiplied
factor out	להוציא גורם משותף	Put a shared factor outside parentheses
coefficient	מקדם	The number multiplying a variable, such as $6$ in $6x$
exponent / power	חזקה	The raised number, such as $3$ in $x^3$
greatest common factor	גורם משותף גדול ביותר	The largest factor shared by every term

A question that says “Factor out the GCF” does not ask you to solve for $x$. It asks you to rewrite the expression as a product.

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How to find the GCF

To find a GCF, examine the numbers and the variables separately.

For the numbers, find the biggest number that divides into every coefficient with no remainder.

For variables, a variable must appear in every term before it can be part of the GCF. When it appears with different exponents, choose the smallest exponent.

Consider:

$$18x^{3}y+24x^2{y}^3+36x^4{y}^2$$

Step 1: Find the GCF of the coefficients

The coefficients are $18$, $24$, and $36$.

The largest number dividing all three is $6$:

$$18÷6=3,24÷6=4,36÷6=6$$

So the numerical part of the GCF is $6$.

Step 2: Find common variables

Look at the powers of $x$:

$$x^3,x^2,x^4$$

Every term contains $x$, and the smallest power is $x^2$. Therefore, take out $x^2$.

Now look at the powers of $y$:

$$y,y^3,y^2$$

Every term contains $y$. The first term has only one $y$, so the smallest power is $y$.

Therefore, the complete GCF is:

$$6x^{2}y$$

A useful rule to memorize:

For each variable shared by all terms, factor out its smallest exponent.

Notice that $n$ is not included in the diagram’s GCF: the first term, $8m^3$, has no $n$. A factor must be present in every term.

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Factoring the GCF out of every term

Once you know the GCF, write it outside parentheses. Then divide each term by that GCF. The answers to those divisions go inside the parentheses.

Using the expression from above:

$$18x^{3}y+24x^2{y}^3+36x^4{y}^2$$

The GCF is $6x^{2}y$, so begin:

$$18x^{3}y+24x^2{y}^3+36x^4{y}^2=6x^{2}y(\text{something})$$

Now divide term by term.

First term:

$$18x^{3}y÷6x^{2}y=3x$$

Second term:

$$24x^2{y}^3÷6x^{2}y=4y^2$$

Third term:

$$36x^4{y}^2÷6x^{2}y=6x^{2}y$$

So the factored expression is:

$$18x^{3}y+24x^2{y}^3+36x^4{y}^2=6x^{2}y(3x+4y^2+6x^{2}y)$$

When dividing powers with the same base, subtract exponents:

$$x^5÷x^3=x^{5-3}=x^2$$

But when the powers are equal, the variable disappears:

$$x^2÷x^2=1$$

That is why a term such as $12x^2÷4x^2$ becomes just $3$, not $3x$.

A shorter example

Factor:

$$16z^5-8z^3+12z$$

The GCF of $16$, $8$, and $12$ is $4$. Every term has $z$, and the smallest exponent is $z^1$, usually written simply as $z$.

So the GCF is:

$$4z$$

Divide every term by $4z$:

$$16z^5÷4z=4z^4$$ $$-8z^3÷4z=-2z^2$$ $$12z÷4z=3$$

Therefore:

$$16z^5-8z^3+12z=4z(4z^4-2z^2+3)$$

Keep the original signs when you divide. The minus sign before $8z^3$ remains a minus sign inside the parentheses.

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Negative signs: a useful factoring choice

A GCF is often written as positive. For example:

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

However, if the first term is negative, it is often useful to factor out a negative common factor. This makes the first term inside the parentheses positive.

Consider:

$$-x^2+400x$$

Both terms contain $x$. You could factor out $x$:

$$x(-x+400)$$

That is correct. But this form is usually more convenient:

$$-x(x-400)$$

Check why:

$$-x(x-400)=-x^2+400x$$

Factoring out a negative changes the signs inside the parentheses because each term is divided by a negative factor.

For example:

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

The shared factor can be written as $-5x$:

$$-10x^3+15x^2-5x=-5x(2x^2-3x+1)$$

Check the middle term carefully:

$$-5x(-3x)=15x^2$$

A negative outside multiplied by a negative inside gives a positive term.

This choice will be especially helpful when you begin factoring quadratic expressions in the next lessons.

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Check factoring by distributing

Factoring should always be checked by distributing — הפעלת חוק הפילוג. If distribution returns the exact original expression, your factorization is correct.

Check:

$$4z(4z^4-2z^2+3)$$

Distribute $4z$ to all three terms:

$$4z\cdot 4z^4=16z^5$$ $$4z\cdot (-2z^2)=-8z^3$$ $$4z\cdot 3=12z$$

So:

$$4z(4z^4-2z^2+3)=16z^5-8z^3+12z$$

This matches the original expression exactly.

Use this short written-test routine:

1. Identify the GCF of the coefficients.
2. Include only variables found in every term.
3. Use the smallest exponent of each common variable.
4. Put the GCF outside parentheses.
5. Divide every term by the GCF.
6. Distribute once to check.

Common mistakes to avoid:

• Taking the largest exponent instead of the smallest one.
• Factoring out a variable that is missing from one term.
• Dividing only the first term instead of every term.
• Losing a negative sign.
• Stopping with a smaller common factor when a larger one exists. For example, $2x$ is common to $12x^2+18x$, but $6x$ is the GCF.

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Key takeaways

• Factoring is the reverse of the distributive property.
• The GCF is the largest factor shared by all terms.
• Find the GCF of the numbers, then include each variable shared by every term with its smallest exponent.
• Write the GCF outside parentheses and divide every original term by it.
• Check by distributing; you must get the original expression back exactly.
• Factoring an expression is different from solving an equation: no value of $x$ is being found.

Next, you will build on this skill by factoring quadratic trinomials of the form $x^2+bx+c$.