Simplifying: Combining Terms

One of the most common operations when working with polynomials is simplifying by combining like terms.  Terms can be combined only if they have exactly the same variable part, for example:

4x and 3 can’t be combined because the second term has no variable.
4x and 3y can’t be combined because they have different variables.
4x² and 3x can’t be combined because they have different exponential values.

It’s like the well known saying that you can’t combine apples and oranges.  Combining like terms follows the same adage.  Here are some examples:

Simplify 3x + 4x

These are like terms since they have the same variables:

3x + 4x = 7x

Simplify 2x² + 3x − 4 − x² + x + 9

It’s usually easier to group like terms together first, then simplify:

2x² + 3x − 4 − x² + x + 9 =
=
(2x² − x²) + (3x + x) + (−4 + 9)
x² + 4x + 5

Some students find it helpful to add the understood coefficient of 1 in front of variable expressions where it is normally omitted, as noted in red below:

2x² + 3x − 4 − x² + x + 9 =
=
=
(2x² − 1x²) + (3x + 1x) + (−4 + 9)
1x² + 4x + 5
x² + 4x + 5

Inserting a one is not required but if it helps you correctly complete the simplification then use it.

Simplify 10x³ − 14x² + 3x − 4x³ + 4x − 6

10x³ − 14x² + 3x − 4x³ + 4x − 6 =
=
(10x³ − 4x³) + (−14x²) + (3x + 4x) − 6
6x³ − 14x² + 7x − 6

This may seem obvious but don’t confuse multiplication and addition, it’s one of the most commonly made mistakes:

(x)(x) = x² (multiplication)
x + x = 2x (addition)
x² DOES NOT equal 2x

So, don’t say x³ + x² = x⁵ or 5x, and don’t say 2x + x = 2x².

This topic is associated with the Order of Operations but simplifying with parentheses is an area that often causes difficulty.  When simplifying expressions with parentheses, you will be applying the Distributive Property to multiply through the parentheses in order to simplify a given expression.  As this lesson progresses, note the importance of simplifying as you go and of doing each step neatly, completely and exactly.

Simplify 3(x + 4).

To simplify this, we have to get rid of the parentheses.  The Distributive Property says to multiply the 3 onto everything inside the parentheses:

3(x + 4) = (3 • x) + (3 • 4) = 3x + 12

A common error at this stage is to take the 3 through the parentheses but only onto the x, forgetting to carry it through to the 4 as well.  If you need to draw arrows to help you remember to carry through onto everything inside the parentheses, then use them!

Simplify −2(x − 4)

We have to take the −2 through the parentheses.  This gives us:

−2(x − 4)
−2(x) − 2(−4)
−2x + 8

A another common mistake is to lose a minus sign, such as doing −2(x − 4) = −2(x) − 2(4) = −2x − 8.  Be careful with the minus signs!  Until you are confident in your skills, take the time to write out the distribution, complete with the signs, as in the example.

If you have difficulty with the subtraction, try converting it to addition of a negative:

−2(x − 4)
−2(x + [−4])
−2(x) + (−2)(−4)
−2x + 8

Do as many steps as you need to, in order to consistently get the correct answer.

Simplify −(x − 3)

In problems like this, many students find it helpful to write in the understood 1 before the parentheses.  Check with your teacher but you should be able to use the technique that works best for you:

−(x − 3)
1(x − 3)
1(x) −1(−3)
1x + 3
x + 3

Note that, technically, −1x + 3 and −x + 3 are the same thing and usually, either should be an acceptable answer.  However, some teachers will accept only −x + 3 and would count −1x + 3 as not fully simplified.  Again, if in doubt, ask your teacher.

Simplify 25 − (x + 3 − x²)

Take the minus through the parentheses:

25 − (x + 3 − x²)
25 −1(x + 3 − x²)
25 − x − 3 + x²
x² − x + 25 − 3
x² − x + 22

Simplify 2 + 4(x − 1)

Remember the order of operations: multiplication comes before addition.  You can’t do the 2 + until you have taken the 4 through the parentheses.

2 + 4(x − 1)
2 + 4(x) + 4(−1)
2 + 4x + (−4)
2 − 4 + 4x
−2 + 4x
4x − 2

Some teachers will accept either 4x − 2 or −2 + 4x as valid answers however, most texts expect answers to be written in descending order, with the variable terms first, then the constants.  You should know that the two expressions of the answer are the same, but some teachers insist that the answer be written in descending order and it would probably be best to get in the habit doing so.

With nested terms, you typically work from the inside out.  Simplification works the same way, but you need to be more careful.

Simplify 4[x + 3(2x + 1)]

First we take the 3 through the inner brackets before dealing with the 4 and the parentheses.  We’ll also simplify as we go and write each step out completely:

4[x + 3(2x + 1)]
4[x + 3(2x) + 3(1)]
4[x + 6x + 3]
4[7x + 3]
4[7x] + 4[3]
28x+ 12

There’s no particular significance to the order in which you use parentheses, brackets [ ] and braces { }.  They’re primarily used to differentiate between nested terms.  The traditional equence of grouping symbols, working from the iside out, is parentheses, then brackets, then braces, repeating the sequence, as necessary, but this is not a rule; it’s just a tradition.

Simplify 9 − 3[x − (3x + 2)] + 4

We won’t do anything with the 9 − or the + 4 until we simplify inside the brackets and parentheses.  Again, work from the inside out:

9 − 3[x − (3x + 2)] + 4
9 − 3[x − 1(3x + 2)] + 4
9 − 3[x − 1(3x) − 1(2)] + 4
9 − 3[x − 3x − 2] + 4
9 − 3[−2x − 2] + 4
9 − 3[−2x) − 3[−2] + 4
9 + 6x + 6 + 4
6x + 19

It is not required that you go through a specific number of steps.  Use as many as you need and be careful to do one step at a time, writing things out completely and simplifying as you go.  The plan is to arrive at the correct answer.

Simplify 5 + 2{ [3 + (2x − 1) + x] − 2}

Once again, work carefully from the inside out:

5 + 2{ [3 + (2x − 1) + x] − 2}
5 + 2{ [3 + 2x − 1 + x] − 2}
5 + 2{ [2x + x + 3 − 1] − 2}
5 + 2{ [3x + 2] − 2}
5 + 2{3x + 2 − 2}
5 + 2{3x}
5 + 6x

Simplify x + 2(x − [3x − 8] + 3)

This is the type of problem you’re likely to see on tests.  This is just an Order of Operations problem with a variable.  Work carefully from the inside out and you should be fine:

x + 2(x − [3x − 8] + 3)
x + 2(x −1[3x − 8] + 3)
x + 2(x − 3x + 8 + 3)
x + 2(−2x + 11)
x − 4x + 22
−3x + 22

Simplify ([6x − 8] − 2x) − ([12x − 7] − [4x − 5])

Again, work from the inside out:

([6x − 8] − 2x) − ([12x − 7] − [4x − 5])
(6x − 8 − 2x) − (12x − 7 − 4x + 5)
(4x − 8) −1(8x − 2)
4x − 8 − 8x + 2
−4x − 6

Simplify −4y − (3x + [3y − 2x + {2y − 7} ] − 4x + 5)

−4y − (3x + [3y − 2x + {2y − 7}] – 4x + 5)
−4y − (3x + [3y − 2x + 2y − 7] – 4x + 5)
−4y − (3x + [−2x + 5y − 7] − 4x + 5)
−4y − (3x − 2x + 5y − 7 − 4x + 5)
−4y − (3x − 2x − 4x + 5y − 7 + 5)
−4y −1(−3x + 5y − 2)
−4y + 3x − 5y + 2
3x − 4y − 5y + 2
3x − 9y + 2

Related to simplification problems are equations and there’s a notable difference.  With simplification, you’re reducing an algebraic expression to its lowest terms.  An equation however, is mathematical statement that has two expressions separated by an equal sign.  The expression on the left of the equal sign has the same value as the expression on the right.  One or both of the expressions may contain variables and solving an equation means manipulating the expressions to find the value of the variables.  For instance:

Solve 3 + 2[4x − (4 + 3x)] = −1

As usual, we work from the inside out:

3 + 2[4x − (4 + 3x)] =
3 + 2[4x − 1(4 + 3x)] =
3 + 2[4x − 1(4) − 1(3x)] =
3 + 2[4x − 4 − 3x] =
3 + 2[1x − 4] =
3 + 2[1x] + 2[−4] =
3 + 2x − 8 =
2x + 3 − 8 =
2x − 5 =
2x − 5 + 5 =
2x =
x =
−1
−1
−1
−1
−1
−1
−1
−1
−1
−1 + 5
4
2

There’s no rule concerning how many steps you take in solving (or simplifying) and once you become expert at the process, you may do a lot of this in your head, but until you reach that level, you should write things out clearly and completely.  Remember that you can check the answers to equations by plugging the solution back into the original equation.  In this case:

Evaluate 3 + 2[4x − (4 + 3x)] for x = 2

3 + 2[4(2) − (4 + 3(2))]
3 + 2[8 − (4 + 6)]
3 + 2[8 − (10)]
3 + 2[−2]
3 − 4
−1

So, 3 + 2[4x − (4 + 3x)] = −1 and since this matches the original equation, we know that x = 2 is the correct solution.

Note the difference between these two procedures.  The first was an equation requiring a solution whereas thesecond simply evaluated the equation to verify the result.  Evaluation and simplification do not require a solution but an equation MUST reach a conclusion!  Be careful that you don’t make the mistake of converting a simplification problem into an equation.

Simplify 4 − 10[x + (2x − 3)] + 12x

In addition to the instructions, you know this is a simplification problem because there is no equal sign in the expression.  We’ll simplify, from the inside out, and end up with an equivalent expression.

4 − 10[x + (2x − 3)] + 12x
4 − 10[x + 2x − 3] + 12x
4 − 10[3x − 3] + 12x
4 − 10[3x] − 10[−3] + 12x
4 − 30x + 30 + 12x
−30x + 12x + 4 + 30
−18x + 34

Solve 2(x + 3) = 4 − (2 − x)

This is an equation since there is an equal sign in the original problem…DON’T LOSE IT!

2(x + 3) =
2(x) + 2(3) =
2x + 6 =
2x + 6 =
x − x + 6 =
x + 6 =
x + 6 − 6 =
x =
 4 − (2 − x)
4 − 1(2) − 1(−x)
4 − 2 + 1x
2 + x
2 + x − x
2
2 − 6
−4

To stress once again: you need to take the time to write out each step, as many as you need.  Work from the inside out and be careful with the minus signs.  Don’t forget the Order of Operations, and don’t make the mistake of confusing simplifying with solving.

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