Factoring and Recognizing Equivalent Linear Expressions | Grade 8 Math | LevelKite

This lesson builds on a move you already know. Throughout it, the variable $x$ stands for any real number, which means any number on the number line.

You know how to multiply with the distributive property: $3(2x + 5)$ becomes $6x + 15$ . Today you run that move backward. Running it backward is called factoring. Then you learn how to decide whether two expressions that look different always have the same value.

Every expression here is a linear expression: you can write it as a number times $x$ plus a number. The parts that are added or subtracted are its terms, such as $6x$ and $15$ in $6x + 15$ .

\underbrace{3(2x + 5)}_{\text{factored form}} = \underbrace{6x + 15}_{\text{expanded form}}

Distributing takes you from the factored form, which is a product, to the expanded form, which is a sum. Factoring goes the other way.

Either way, the new expression must be equivalent to the old one. Two expressions are equivalent when they have the same value for every permitted value of the variable. A value is permitted when you can compute the expression there. Linear expressions use only multiplying by numbers, adding and subtracting, and you can always do those. So here every real number is permitted.

The table tests both forms at four values of $x$ .

Both forms give the same value at every tested number

Tested value	Factored form	Expanded form
$x = -2$	$3(2(-2) + 5) = 3(-4 + 5) = 3(1) = 3$	$6(-2) + 15 = -12 + 15 = 3$
$x = 0$	$3(2(0) + 5) = 3(0 + 5) = 3(5) = 15$	$6(0) + 15 = 0 + 15 = 15$
$x = \tfrac{1}{2}$	$3(2 \cdot \tfrac{1}{2} + 5) = 3(1 + 5) = 3(6) = 18$	$6 \cdot \tfrac{1}{2} + 15 = 3 + 15 = 18$
$x = 4$	$3(2(4) + 5) = 3(8 + 5) = 3(13) = 39$	$6(4) + 15 = 24 + 15 = 39$

Key concept

Factoring is distributing in reverse

A factor is one of the numbers or expressions multiplied together in a product. The distributive property says that $a(b + c)$ and $ab + ac$ have the same value for all real numbers $a$ , $b$ and $c$ . Read left to right, the formula distributes. Read right to left, it factors: it takes the shared factor $a$ out of both terms and writes a sum as a product.

The table could test only four numbers. The property covers every real number. So a factored form and its expanded form are equivalent: they have the same value for every permitted value of the variable. For linear expressions, that is every real number.

a(b + c) = ab + ac

To factor a linear expression, you write it as a number times a simpler linear expression. Here the coefficient (the number multiplying $x$ ) and the constant (the term with no variable) are integers, which are the whole numbers and their opposites. So you can look for a factor they share.

A common factor of two integers is an integer that divides both with no remainder. The greatest common factor (GCF) is the largest one.

Find the GCF of the coefficient and the constant. Ignore negative signs while you search.
Divide each term by the GCF. Each quotient tells you what to multiply the GCF by to rebuild that term. The GCF is at least $1$ , so you never divide by zero.
Write the GCF outside parentheses and the quotients inside.
Check by distributing. You must get the original expression back.

An expression is factored completely when the numbers left inside the parentheses share no factor greater than $1$ . Taking out the GCF does that. If the GCF is $1$ , as in $3x + 5$ , there is no whole number greater than $1$ to take out.

Worked example

Take out the greatest common factor

Factor $12x + 18$ completely.

Find the greatest common factor of the coefficient 12 and the constant 18. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18. The largest number on both lists is 6.
$\text{GCF of } 12 \text{ and } 18 \text{ is } 6$
Divide each term by 6 to find what 6 must be multiplied by to rebuild that term. You may divide because 6 is not zero.
$12x \div 6 = 2x \qquad 18 \div 6 = 3$
Rewrite each term as 6 times its quotient. Each product equals the term it replaces, so the value stays the same for every value of the variable.
$12x + 18 = 6 \cdot 2x + 6 \cdot 3$
Both terms now show the factor 6. Use the distributive property in reverse to write 6 once, outside the parentheses.
$6 \cdot 2x + 6 \cdot 3 = 6(2x + 3)$
Check by distributing. You get the original expression back. Inside the parentheses, 2 and 3 share no factor greater than 1, so the factoring is complete.
$6(2x + 3) = 6 \cdot 2x + 6 \cdot 3 = 12x + 18$

Answer

Factored completely, $12x + 18$ is $6(2x + 3)$ .

Worked example

Take out a negative factor

Factor $-8x + 20$ completely so that the coefficient of $x$ inside the parentheses is positive.

Find the greatest common factor, ignoring the negative sign for now. The factors of 8 are 1, 2, 4 and 8. The factors of 20 are 1, 2, 4, 5, 10 and 20. The largest number on both lists is 4.
$\text{GCF of } 8 \text{ and } 20 \text{ is } 4$
The coefficient of the variable term is negative, so take out negative 4 instead of 4. Negative 4 is also a common factor and it is not zero, so you may divide each term by it. Negative 8 and negative 4 have the same sign, so their quotient is positive. The numbers 20 and negative 4 have different signs, so their quotient is negative.
$-8x \div (-4) = 2x \qquad 20 \div (-4) = -5$
Rewrite each term as negative 4 times its quotient. Each product equals the term it replaces, so the value is unchanged.
$-8x + 20 = (-4)(2x) + (-4)(-5)$
Use the distributive property in reverse to write negative 4 once, outside the parentheses. Adding negative 5 is the same as subtracting 5.
$(-4)(2x) + (-4)(-5) = -4\big(2x + (-5)\big) = -4(2x - 5)$
Check by distributing and watch the signs. Negative 4 times 2 is negative 8, and negative 4 times negative 5 is positive 20. Inside the parentheses, 2 and 5 share no factor greater than 1, so the factoring is complete.
$-4(2x - 5) = (-4)(2x) + (-4)(-5) = -8x + 20$

Answer

With a positive coefficient of $x$ inside the parentheses, $-8x + 20$ factors completely as $-4(2x - 5)$ .

Now for the second skill. You could also factor $-8x + 20$ as $4(-2x + 5)$ . That looks different from $-4(2x - 5)$ , yet both distribute back to $-8x + 20$ , so all three expressions are equivalent. You cannot judge equivalence by looks.

To decide whether two linear expressions are equivalent, rewrite each one in simplest form: one $x$ -term and one constant, like $18x - 8$ . A missing constant counts as $0$ . You get there by distributing and by combining like terms (terms with the same variable part). Both moves keep the value the same for every real number $x$ . Then compare:

Same coefficient of $x$ and same constant: equivalent, because both expressions equal that one simplest form for every real number $x$ .
Different constants: not equivalent. At $x = 0$ each simplest form equals its constant, so the values differ there.
Same constant but different coefficients of $x$ : not equivalent. At $x = 1$ each simplest form equals its coefficient plus its constant, so the values differ there.

Worked example

Sort equivalent from not equivalent

Which of these are equivalent to $4(3x - 2) + 6x$ ? Expression A is $2(9x - 4)$ . Expression B is $18x - 2$ .

Put the given expression in simplest form. Start by distributing the 4 to both terms inside the parentheses. Because the 2 is subtracted, its product with 4 is subtracted too.
$4(3x - 2) + 6x = 4 \cdot 3x - 4 \cdot 2 + 6x = 12x - 8 + 6x$
Move the two variable terms next to each other, keeping the minus sign with the 8, and add them. Reordering the terms of a sum does not change its value.
$12x - 8 + 6x = 12x + 6x - 8 = 18x - 8$
Put Expression A in simplest form by distributing the 2. Its coefficient is 18 and its constant is negative 8. Both match the given expression, so A is equivalent to it. Going the other way, taking the greatest common factor 2 out of the simplest form gives Expression A back.
$2(9x - 4) = 2 \cdot 9x - 2 \cdot 4 = 18x - 8$
Expression B is already in simplest form. Its coefficient is 18, which matches, but its constant is negative 2 instead of negative 8. Different constants mean B is not equivalent.
$\text{Given: } 18x - 8 \qquad \text{B: } 18x - 2$
Confirm by substituting 0 for the variable in the original expressions. The two values are different, and one mismatch is enough to show that B is not equivalent.
$\text{Given: } 4(3 \cdot 0 - 2) + 6 \cdot 0 = 4(-2) + 0 = -8 \qquad \text{B: } 18 \cdot 0 - 2 = 0 - 2 = -2$

Answer

Expression A, $2(9x - 4)$ , is equivalent to $4(3x - 2) + 6x$ because both simplify to $18x - 8$ . Expression B, $18x - 2$ , is not equivalent.

Takeaway. Factoring runs the distributive property in reverse. You divide every term by a common factor that is not zero, usually the greatest one, write that factor outside the parentheses, and distribute to check. The factored form is equivalent to the original: the two have the same value for every real number $x$ . To decide whether any two linear expressions are equivalent, compare their simplest forms. They must have the same coefficient of $x$ and the same constant. One mismatched value rules equivalence out, but one matching value is not enough to prove it.