Last modified on December 18th, 2023

chapter outline

 

Simplifying Algebraic Expressions

Simplifying algebraic expressions means rewriting the algebraic expressions in the simplest form with no like terms and without any operators like addition, subtraction, multiplication, and division.

By Combining Like Terms

The algebraic expressions involving the variables with whole number coefficients (no fractions, radicals, etc.) can be simplified by combining and solving the like terms (terms with the same variables and exponents). Then, the expression will only be left with the unlike terms that can not be simplified further.

For example,

Let us simplify the expression 15x – 1 – 3x + 8

Now, identifying the like terms, we get 15x and -3x, -1 and 8 as two pairs of the like terms.

Then, by combining them, we get 

= (15x – 3x) + (-1 + 8)

Solving them, we get, 

= 12x + 7

Thus, 15x – 1 – 3x + 8 can be simplified as 12x + 7.

By Order of Operations

To simplify the above algebraic expression, we follow the general order of operations PEMDAS – which stands for 

P – Parentheses, E – Exponents, M – Multiplication, D – Division, A – Addition, and S – Subtraction.

Parentheses

Now let us simplify the algebraic expression ${\left( x\times x^{2}\right) -\left( 4x\div 2\right) +3\times 9x-4\div 2}$

First, we solve the parentheses by dividing the terms inside the bracket; we get

= ${\left( x\times x^{2}\right) -2x+3\times 9x-4\div 2}$

Exponent

Now, simplifying the terms containing exponents by the exponent rule (${a^{m}\div a^{n}=a^{m-n}}$, when ${a\neq 0}$), we get,

= ${x^{3}-2x+3\times 9x-4\div 2}$

Multiplication and Division

Next, we multiply and then divide according to the order of operations, and we get

= ${x^{3}-2x+27x-4\div 2}$

= ${x^{3}-2x+27x-2}$

Addition and Subtraction

Now, by combining the like terms, adding or subtracting them, and writing the unlike terms as they are, we get

= ${x^{3}+25x-2}$

While simplifying an expression, the result must be in the standard form (from the highest to the lowest power).

Thus, the given expression ${\left( x\times x^{2}\right) -\left( 4x\div 2\right) +3\times 9x-4\div 2}$ is simplified to ${x^{3}+25x-2}$.

Sometimes, if we have a variable and a number inside the bracket, then, using parentheses, we can not solve the terms inside the bracket. Then, we need to expand the expressions.

Expanding Using Distributive Property

The distributive property states that an expression given in the form of: 

x(y + z) can be simplified as xy + xz 

Or, x(y – z) can be simplified as xy – xz. 

Let us expand and simplify the expression 2(x + 4 + 3x) + 3(x – 5 + 7) – 2y 

First, we combine the like terms,

2{(x + 3x) + 4} + 3{x + (- 5 + 7)} – 2y

= 2(4x + 4) + 3(x + 2) – 2y

Now, using the distributive property, we get,

2(4x) + (2 x 4) + 3(x) + (3 x 2) – 2y

= 8x + 8 + 3x + 6 – 2y

Finally, by combining the like terms, we get

(8x + 3x) – 2y + (8 + 6)

= 11x – 2y + 14

Thus, 2(x + 4 + 3x) + 3(x – 5 + 7) – 2y can be simplified as 11x – 2y + 14.

Simplifying with Fractions

If fractions are given in any expression, then we use the exponent rules and the distributive property to simplify such expressions. 

 Let us simplify ${\dfrac{5}{2}\left( 2x+8\right) +\dfrac{y}{4}\left( 4+x\right)}$

Using distributive property, we get,

${\left( \dfrac{5}{2}\times 2x\right) +\left( \dfrac{5}{2}\times 8\right) +\left( \dfrac{y}{4}\times 4\right) +\left( \dfrac{y}{4}\times x\right)}$

= ${5x+\dfrac{xy}{4}+y+20}$

Thus, ${\dfrac{5}{2}\left( 2x+8\right) +\dfrac{y}{4}\left( 4+x\right)}$ can be simplified as ${5x+\dfrac{xy}{4}+y+20}$.

While simplifying an algebraic expression with a fraction, the fraction must be in the simplest form, and only the unlike terms are kept using any change.

Simplifying with Factors

Some expressions require factoring for simplification. In such an expression, we remove the common factors among all the terms and keep the remaining ones unchanged. 

Now, let us simplify the expression ${\dfrac{2}{5}\left( 10x^{2}+5x-35\right)}$

= ${\dfrac{2\times 5\left( 2x^{2}+x-7\right) }{5}}$

= ${2\left( 2x^{2}+x-7\right)}$

= ${4x^{2}+2x-14}$

Thus, the simplified algebraic expression is ${4x^{2}+2x-14}$.

However, we must remember some other rules to simplify an algebraic expression.

Other Rules

  • To add/subtract the like terms, we add/subtract the coefficients of those terms and write the common variable with it.
  • If there is a negative sign just before the parentheses, we reverse the signs of the terms inside the brackets.
  • If there is a positive sign just before the parentheses, we remove the brackets and keep the signs of the terms unchanged.

Solved Examples

Simplify the following expressions:
a) 5x – ${\left( -2x^{2}+3x-1\right)}$
b) ${4ab-2b+3\left( ab+1\right) -2b}$
c) ${\dfrac{5x^{2}}{10x^{2}+5x^{3}}}$
d) ${\dfrac{1}{2}\left( 10x^{2}-34\right)}$

Solution:

a) The given algebraic expression is 5x – ${\left( -2x^{2}+3x-1\right)}$
Solving the parentheses, we get,
5x + ${2x^{2}}$ – 3x + 1
Now, by combining all the like terms, we get,
${2x^{2}}$ + (5x – 3x) + 1
= ${2x^{2}}$ + 2x + 1
Thus, the simplified algebraic expression is ${2x^{2}}$ + 2x + 1.
b) The given algebraic expression is ${4ab-2b+3\left( ab+1\right) -2b}$
Solving the parentheses, we get,
4ab – 2b + 3ab + 3 – 2b
Now, by combining all the like terms, we get,
(4ab + 3ab) + (-2b – 2b) + 3
= 7ab – 4b + 3
Thus, the simplified algebraic expression is 7ab – 4b + 3.
c) The given algebraic expression is${\dfrac{5x^{2}}{10x^{2}+5x^{3}}}$
By finding the common factors, we get,
${\dfrac{5x^{2}}{5x^{2}\left( 2+x\right) }}$
= ${\dfrac{1}{\left( 2+x\right) }}$
Thus, the simplified algebraic expression is ${\dfrac{1}{\left( 2+x\right) }}$.
d) The given algebraic expression is ${\dfrac{1}{2}\left( 10x^{2}-34\right)}$
By finding the common factors, we get,
${\dfrac{2\left( 5x^{2}-17\right) }{2}}$
= ${5x^{2}-17}$
Thus, the simplified algebraic expression is ${5x^{2}-17}$.

Simplify the algebraic expression 15xy-13+4x+3y+xy+21

Solution:

The given algebraic expression is 15xy-13+4x+3y+xy+21
By combining all the like terms, we get,
4x + (15xy + xy) + 3y + (21 – 13)
= 4x + 16xy + 3y + 8
Thus, the simplified algebraic expression is 4x + 16xy + 3y + 8.

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