2.2: Simplifying Algebraic Expressions
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- Apply the distributive property to simplify an algebraic expression.
- Identify and combine like terms.
Distributive Property
The properties of real numbers are important in our study of algebra because a variable is simply a letter that represents a real number. In particular, the distributive property states that given any real numbers
This property is applied when simplifying algebraic expressions. To demonstrate how it is used, we simplify
Certainly, if the contents of the parentheses can be simplified, do that first. On the other hand, when the contents of parentheses cannot be simplified, multiply every term within the parentheses by the factor outside of the parentheses using the distributive property. Applying the distributive property allows you to multiply and remove the parentheses.
Example
Simplify:
Solution:
Multiply
Answer:
Simplify:
Solution:
Multiply
Answer:
Simplify:
Solution:
Apply the distributive property by multiplying only the terms grouped within the parentheses by
Figure
Answer:
Because multiplication is commutative, we can also write the distributive property in the following manner:
Simplify:
Solution:
Multiply each term within the parentheses by
Answer:
Division in algebra is often indicated using the fraction bar rather than with the symbol (
Rewriting algebraic expressions as products allows us to apply the distributive property.
Divide:
\(\frac{25x^{2}-5x+10}{5}.
Solution:
First, treat this as
Alternate Solution:
Think of
Answer:
We will discuss the division of algebraic expressions in more detail as we progress through the course.
Simplify:
- Answer
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Combining Like Terms
Terms with the same variable parts are called like terms, or similar terms. Furthermore, constant terms are considered to be like terms. If an algebraic expression contains like terms, apply the distributive property as follows:
In other words, if the variable parts of terms are exactly the same, then we may add or subtract the coefficients to obtain the coefficient of a single term with the same variable part. This process is called combining like terms. For example,
Notice that the variable factors and their exponents do not change. Combining like terms in this manner, so that the expression contains no other similar terms, is called simplifying the expression. Use this idea to simplify algebraic expressions with multiple like terms.
Simplify:
Solution:
Identify the like terms and combine them.
Answer:
In the previous example, rearranging the terms is typically performed mentally and is not shown in the presentation of the solution.
Simplify:
Solution:
Identify the like terms and add the corresponding coefficients.
Answer:
Simplify:
Solution:
Remember to leave the variable factors and their exponents unchanged in the resulting combined term.
Answer:
Simplify:
To add the fractional coefficients, use equivalent coefficients with common denominators for each like term.
Answer:
Simplify:
Solution:
Consider the variable part to be
Answer:
Simplify:
- Answer
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Distributive Property and Like Terms
When simplifying, we will often have to combine like terms after we apply the distributive property. This step is consistent with the order of operations: multiplication before addition.
Simplify:
Solution:
Distribute
Figure
Answer:
In the example above, it is important to point out that you can remove the parentheses and collect like terms because you multiply the second quantity by
Simplify:
- Answer
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Often we will encounter algebraic expressions like
This leads to two useful properties,
Simplify:
Solution:
Multiply each term within the parentheses by
Figure
Answer:
When distributing a negative number, all of the signs within the parentheses will change. Note that
Simplify:
Solution:
The order of operations requires that we multiply before subtracting. Therefore, distribute
Answer:
It is worth repeating that you must follow the order of operations: multiply and divide before adding and subtracting!
Simplify:
- Answer
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Subtract
Solution:
First, group each expression and treat each as a quantity:
Next, identify the key words and translate them into a mathematical expression.
Figure
Finally, simplify the resulting expression.
Answer:
Key Takeaways
- The properties of real numbers apply to algebraic expressions, because variables are simply representations of unknown real numbers.
- Combine like terms, or terms with the same variable part, to simplify expressions.
- Use the distributive property when multiplying grouped algebraic expressions,
. - It is a best practice to apply the distributive property only when the expression within the grouping is completely simplified.
- After applying the distributive property, eliminate the parentheses and then combine any like terms.
- Always use the order of operations when simplifying.
Multiply.
- Answer
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Translate the following sentences into algebraic expressions and then simplify.
- Simplify two times the expression
. - Simplify the opposite of the expression
. - Simplify the product of
and . - Simplify the product of
and .
- Answer
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Simplify.
- Answer
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Simplify.
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Translate the following sentences into algebraic expressions and then simplify.
- What is the difference of
and ? - Subtract
from . - Subtract
from twice the quantity . - Subtract three times the quantity
from .
- Answer
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- Do we need a distributive property for division,
? Explain. - Do we need a separate distributive property for three terms,
? Explain. - Explain how to subtract one expression from another. Give some examples and demonstrate the importance of the order in which subtraction is performed.
- Given the algebraic expression
, explain why subtracting is not the first step. - Can you apply the distributive property to the expression
? Explain why or why not and give some examples. - How can you check to see if you have simplified an expression correctly? Give some examples.
- Answer
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