GED Mathematical Reasoning: Simplifying Algebraic Expressions

  • Simplifying an algebraic expression means to perform all the operations possible. Part of this process involves what we call “combining like terms.”
  • A term is number, a variable, or the product or quotient of numbers and variables. Here are some examples of terms.

3x
-2y^2
10
b^3
\frac{a}{5}

  • “Like terms” are terms that have the same variable part. Meaning, the variables are the same and the exponents are the same.
  • 8 “x” and  -14 “x” squared are NOT considered “like terms.”Although they both have an “x” one has an exponent of 2 and the other does not.

 

Combining like terms

  • First of all, we usually start with the terms containing the highest exponent. So let’s begin with -14 “x” squared and -2 “x” squared.
  • To combine these like terms, add the number part and keep the variable part the same.

As a side note, having the same terms but in a different order isn’t technically incorrect. But as stated, it is customary to put the terms in decreasing order starting with the term containing the highest exponent. It is also customary to put the terms in alphabetical order if more than one letter is used. As we will see in example two.

 

Distributive property

  • The distributive property is used to remove parenthesis from an expression so that further simplification can be accomplished. It’s used when one term is multiplied by a sum or difference in parenthesis. It doesn’t matter if there are two or more terms being added or subtracted inside the parenthesis.
  • By the distributive property, the term outside parenthesis is multiplied by each term inside the parenthesis and ONLY the terms within that set of parenthesis. Then the result is added or subtracted, depending on the signs and as “like terms” will allow.
  • Formally, the distributive property is stated as follows:

 a(b + c) = ab + ac

Or

 a(b - c) = ab - ac

 

Example 1

Simplify: 8x + 3 - 14x^2 - 7x - 2x^2 + 10

 

Notice that the expression in example one contains six terms. We can think of the preceding subtraction signs as negative signs, and list the terms as follows:

8x
4
-14x^2
-7x
-2x^2
10

Now that we’ve discussed what terms are, let’s talk about “like terms.” Looking again at example one, notice that there are three matching pairs of “like terms”:

8 “x” and  -7 “x”

4 and 10

-14 “x” squared and -2 “x” squared

Although the numbers in front are different, each pairing has the same variable and the same exponent. Let’s combine like terms by adding the number part and keep the variable part the same.

We usually start with the terms containing the highest exponent. So let’s begin with -14 “x” squared and -2 “x” squared.

 - 14x^2 + -2x^2 = (-14 + -2)x^2 = - 16x^2

So to combine -14 “x” squared and -2 “x” squared, we’ll first add -14 and -2, which is equal to -16. And keep the variable part, which is “x” squared.

Now let’s combine 8 “x” and  -7 “x”. We’ll add 8 and -7, which is equal to 1. And keep the variable part as “x”.  The result is  1 “x” or just “x”

8x +  -7x = (8 +  -7)x = 1x = x

Finally, we’ll combine the numbers 4 and 10, which is equal to 14.

4+10=14

For the final answer to example one, let’s put the combinations of these terms together to read:  -16 “x” squared plus “x” plus 14. Notice that we precede the positive terms with addition signs.

-16x^2 + x + 14

 

Example 2

Simplify: -3(a + 5) + 2(b - 4)

 

Example 2 contains two sets of parenthesis – one containing a sum, the other containing a difference. However, we can’t add or subtract the terms within the parenthesis because they are not “like terms.”

This is where the distributive property comes in. We can use the distributive property to clear the parenthesis, which will allow us to further simplify the problem by coming like terms. We’ll first distribute the -3 to the first set of parenthesis and ONLY the terms within the first set. Think of the closing parenthesis as a “stop sign” – to tell us when to stop distributing. Some people find it helpful to draw arrows when using the distributive property.

math 174

-3 times “a” equals -3a. -3 times positive 5 equals -15.

Now let’s distribute the 2. 2 times “b” equals 2b. 2 times 4 equals 8.

-3 \cdot a +  -3 \cdot 5 + 2 \cdot b - 2 \cdot 4
-3a - 15 + 2b - 8

And now we can go about combining like terms. The only “like terms” in this case are -15 and -8, which combine to be -23.

-3a + 2b - 23

Before we move on to our final example related to evaluating an algebraic expression, I have one more note to mention about the distributive property.

Consider the expression with 3x plus 9 in parenthesis, preceded by a subtraction or negative sign.

-(3x + 9)

To use the distributive property in this situation, imagine a ONE following the negative sign and distribute the -1.

-(3x + 9)
-1(3x + 9)

math 175

So we have -1 times 3x, which equals -3x. And -1 times 9, which equals -9. So this expression simplifies to: -3x plus -9, or just -3x minus 9.

-1 \cdot 3x +  -1 \cdot 9
-3x +  -9
-3x - 9

Example 3

Evaluate when m = 6 and n = -3:

\frac{m + n^2}{3} + 2m

 

To evaluate an algebraic expression means to substitute the given values and follow the order of operations to find the answer. In this case, we’ll substitute 6 for “m” and -3 for “n”. When you substitute values to evaluate an expression, I recommend putting parenthesis around the numbers. We’ll talk about why in just a minute.

So we have: 6 plus -3 in parenthesis squared, over 3, plus 2 times 6

\frac{(6) + {(-3)}^2}{3} + 2(6)

Now that we have substituted the given values for “m” and “n” we can follow the order of operations to simplify the expression. There are no grouping symbols containing an operation, so we will first evaluate -3 squared. The base is -3 and the exponent is 2, so -3 squared is equal to -3 times -3, which is +9.

\frac{(6) + {(-3 \cdot -3)}}{3} + 2(6)
\frac{(6) + (9)}{3} + 2(6)

Next we will add the numerator.  6 plus 9 is equal to 15.

\frac{15}{3} + 2(6)

At this point, we have three operations: division, addition, and multiplication. Always do multiplication and division before addition or subtraction and in order from left to right. This means we will divide 15 by 3 next, which is equal to 5.

5 + 2(6)

5 + 12 = 17

Before we conclude this session, it’s worth discussing the importance of using parenthesis around substituted values in an expression.

If we hadn’t placed parenthesis around the -3 “squared” in example 3, we may have arrived at an incorrect answer. This is because -3 in parenthesis squared is very different than -3 squared without parenthesis.

{(-3)}^2 versus -3^2

Parenthesis around the -3 act like glue keeping the negative sign WITH the three, making the base -3. That’s why -3 in parenthesis squared is equal to -3 times -3 which is positive 9.

-3 squared without parenthesis doesn’t have “glue” to hold the negative to the 3. So the base in this case is just 3. 3 squared is 9 and the negative out front makes the final answer -9, which is a very different result.

{(-3)}^2 = (-3)(-3) = 9

-3^2 = -(3)(3) = -9

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