Learning to simplify algebraic expressions is an important step in understanding and mastering algebra. Simplification of algebraic expressions is very useful as it allows us to transform complex or long algebraic expressions into a more compact and simple form.
In this article, we will learn some techniques to simplify any algebraic expression.
Revision of terminology
Let’s review some of the most common terms used to simplify algebraic expressions.
- A variable is a letter that represents an unknown value. Some examples of variables are x, a, b, y, t.
- The coefficient is a numerical value used in conjunction with a variable.
- A constant is a term that has a definite value.
- Like terms are variables with the same letter and power. Like terms can sometimes contain different coefficients. For example, $latex 3 {{x}^2}$ and $latex 5 {{x}^2}$ are like terms since they both have the same variable with a similar exponent. Similarly, $latex 3xy$ and $latex 4xz$ are not like terms since they do not have the same variables.
How to simplify algebraic expressions?
Simplifying algebraic expressions can be defined as the process of writing an expression in the simplest and most compact form possible without affecting the value of the original expression.
To simplify algebraic expressions, we can follow the following simple steps and rules:
- Remove any grouping symbols like parentheses and brackets when multiplying factors.
- Combine like terms using addition and subtraction.
- Combine the constants.
We are going to look at the three most important cases of simplification: distributive property, combining like terms, and distributive property with like terms.
Distributive property
The distributive property indicates that for any real numbers a, b, c, we have:
This property is used to simplify algebraic expressions. By applying the distributive property, we can remove the parentheses.
EXAMPLES
- Simplify $latex 4(x+1)$.
Solution: Multiply each term inside the parentheses by 4:
$latex 4\left( {x+1} \right)=4\times x+4\times 1$
$latex =4x+4$
- Simplify $latex -2\left( {2{{x}^{2}}+3x+4} \right)$.
Solution: Multiply each term inside the parentheses by -2:
$$-2\left( {2{{x}^{2}}+3x+4} \right)=-2\times 2{{x}^{2}}-2\times 3x-2\times 4$$
$latex =-4{{x}^{2}}-6x-8$
- Find the simplest version of $latex 2\left( {2a+3b} \right)-2c$.
Solution: Multiply each term inside the parentheses by 2:
$$2\left( {2a+3b} \right)-2c=2\times 2a+2\times 3b-2c$$
$latex =4a+6b-2c$
- Simplify $latex \left( {4x-2y+3} \right)\times 2$.
Solution: Multiply each term inside the parentheses by 2:
$$\left( {4x-2y+3} \right)\times 2=4x\times 2-2y\times 2+3\times 2$$
$latex =8x-4y+6$
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Combine like terms
Terms with the same variable are called like terms. Furthermore, constant terms are also considered like terms.
If the variables of the terms are exactly the same, then we add or subtract the coefficients to obtain a single coefficient with those variables.
EXAMPLES
- Simplify $latex 2a+4b-3a+2b$.
Solution: Identify and combine like terms:
$latex 2a+4b-3a+2b=-1a+6b$
$latex =-a+6b$
- Simplify $latex 2{{x}^{2}}+3x+4+4{{x}^{2}}-2x+3$.
Solution: Identify and combine like terms:
$$2{{x}^{2}}+3x+4+4{{x}^{2}}-2x+3=6{{x}^{2}}+x+7$$
- Simplify $latex 2{{x}^{2}}y-3x{{y}^{2}}+6x{{y}^{2}}-{{x}^{2}}y$.
Solution: Identify and combine like terms:
$$2{{x}^{2}}y-3x{{y}^{2}}+6x{{y}^{2}}-{{x}^{2}}y={{x}^{2}}y+3x{{y}^{2}}$$
- Simplify $latex 10x{{\left( {x+y} \right)}^{2}}-5x{{\left( {x+y} \right)}^{2}}$.
Solution: Identify and combine like terms. We can see that the terms have $latex x{{\left( {x+y} \right)}^{2}}$ in common. Therefore, we can comine 10 and -5:
$$10x{{\left( {x+y} \right)}^{2}}-5x{{\left( {x+y} \right)}^{2}}=5x{{\left( {x+y} \right)}^{2}}$$
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Distributive property and like terms
Many times we will have to combine like terms after applying the distributive property. This is consistent with the order of operations: multiplication before addition.
EXAMPLES
- Simplify $latex 2(3a+4b)-3(a+b)$.
Solution: We have to distribute the 2 and the -3. Then, we combine like terms:
$latex 2(3a+4b)-3(a+b)$
$$=2\times 3a+2\times 4b-3\times a-3\times b$$
$latex =6a+8b-3a-3b$
$latex =3a+5b$
- Simplify $latex 2x-\left( {3{{x}^{2}}-4x+5} \right)$.
Solution: Let’s distribute the -1 and let’s combine like terms:
$$=2x+\left( {-1} \right)\left( {3{{x}^{2}}} \right)+\left( {-1} \right)\left( {-4x} \right)+\left( {-1} \right)\left( 5 \right)$$
$latex =2x-3{{x}^{2}}+4x-5$
$latex =-3{{x}^{2}}+6x-5$
- Simplify $latex 3-2\left( {{{x}^{2}}-2x+5} \right)$.
Solution: The order of operations requires that we multiply before subtracting. So, we distribute the -2 and then we combine like terms:
$$=3+\left( {-2} \right)\left( {{{x}^{2}}} \right)+\left( {-2} \right)\left( {-2x} \right)+\left( {-2} \right)\left( 5 \right)$$
$latex =3-2{{x}^{2}}+4x-10$
$latex =-2{{x}^{2}}+4x-7$
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Key takeaways
- The properties of real numbers apply to algebraic expressions since variables are simply representations of unknown real numbers.
- To simplify expressions, we combine like terms or terms with the same variables.
- Use the distributive property to multiply grouped algebraic expressions.
- It is easier to apply the distributive property only when the expression within the parentheses or group is completely simplified.
- After applying the distributive property, remove the parentheses and then combine like terms.
- Always use the order of operations when simplifying.
See also
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