Last modified on January 15th, 2024

chapter outline

 

Algebraic Expression

Algebraic expression, or variable expression, is a mathematical expression consisting of two main parts, variables and constants, joined together using mathematical operators addition, subtraction, multiplication, division, and exponentiation.

Here are a few examples of algebraic expressions:

  • 2x + 9
  • 5x + 7y + 3z – 11
  • ${5x^{2}+4y-3}$

Parts

Let us consider the algebraic expression 2x + 9 to understand its parts:

Algebraic Expression

Here,

  • x is a variable (an unknown value) 
  • 9 is a constant (a fixed value)
  • 2x and 9 are the terms, and
  • 2 is a coefficient of x

Variables

A symbol whose value is unknown to us is called a variable. It is denoted by small letters and can take any value. 

a, b, and x are a few examples of variables.

Constants

A symbol with a fixed, definite numerical value is called a constant. 

1, 2, and 50 are constants.

Coefficients

The number multiplied by the variable is called its coefficient.

Let us consider the expression 5y + z.

Here, the coefficient of y is 5, and the coefficient of z is 1.

Terms

A term can be a variable alone, a number, a product of two or more variables, or a product of a number and a variable. These numbers or variables are called the factors of a term. 

Thus, a term or a group of terms form an expression, and a term is formed by multiplying its factors.

An example of a term is ${13xy^{2}}$, where 13, x, y, y are factors.

  1. Like Terms: Terms containing the same variables with the same exponents are called like terms. 2x and 3x, 4y and -9y are some examples of like terms. 
  2. Unlike Terms: Terms containing the same variables with different exponents or containing different variables are called unlike terms. 2x and 3xy, 2x and ${5x^{2}}$ are some examples of unlike terms.

In an algebraic expression, terms can be of 2 types:

Types

Based on the number of unlike terms, the algebraic expressions are classified into the following types:

Monomial 

An algebraic expression consisting of only one unlike term is called a monomial expression.

Examples of some monomial expressions are 3x, 4xyz, and ${2x^{2}}$.

Binomial 

An algebraic expression with two unlike terms is known as a binomial expression.

Examples of some binomial expressions are 2x + y, 4z + 7, and ${10x^{2}+5x^{3}}$.

Trinomial 

A trinomial expression is an expression with three unlike terms.

Examples of some trinomial expressions are ${7x^{2}+3x+5}$, – 6x + 3 + 3y, and 5x + 7y + 3z.

Binomial and trinomial expressions are also known as multinomial expressions (an expression with more than one unlike term).

Polynomial

An expression having one or more than one unlike terms with positive integral exponents of variables is known as a polynomial expression. 

Some examples of polynomial expressions are 2x, 3x + 11, x + y + z – 7, etc.

Other Types

Numeric Expression

A numeric expression containing only numbers and operations but never has any variable.

Some examples of numeric expressions are 5, 50, 5 + 39, and 17 x 14.

Equivalent Expressions

Two expressions are equivalent if they are equal in value but look different.

Some examples of equivalent expressions are 5x – 7y and – 7y + 5x, 3a + b and b + 3a. 

Now, let us understand the operations of the algebraic expressions in detail.

Simplifying

To simplify an algebraic expression, first, we must combine the like terms and then perform the basic mathematical operations (addition, subtraction, multiplication, and division) following the rule of PEMDAS. 

Combining Like Terms

All the like terms are combined and solved to obtain a simplified algebraic expression.

Let us consider 2x + 4y – 6x + 3y + ${3xy^{2}}$ – ${8xy^{2}}$ and reduce it to its lowest form to simplify.

Grouping the like terms we get, 

(2x – 6x) + (4y + 3y) + (${3xy^{2}}$ – ${8xy^{2}}$)

To further simplify, we need to add or subtract the like terms.

Adding and Subtracting

To simplify the above algebraic expression further, we add or subtract all the like terms.

For simplifying, we get, (2x – 6x) + (4y + 3y) + (${3xy^{2}}$ – ${8xy^{2}}$)

Now, adding the expressions, we get,

(2x – 6x) + 7y + (${3xy^{2}}$ – ${8xy^{2}}$)

And subtracting the expressions, we get,

-4x + 7y – ${5xy^{2}}$

Thus, 2x + 4y – 6x + 3y + ${3xy^{2}}$ – ${8xy^{2}}$ can be simplified as -4x + 7y – ${5xy^{2}}$.

Sometimes, an expression can also have multiplication and division operators. Then, we need to multiply and divide to simplify further.

Multiplying and Dividing

Every term of the first expression is multiplied by every term of the second to multiply the algebraic expressions.

Whereas for dividing the algebraic expressions, first, we factor the numerator and denominator, cancel all the possible terms, and then simplify the rest.

Let us take the algebraic expressions as 2x + 10 and ${\dfrac{x-4}{2}}$

Now, multiplying the expressions, we get,

(2x + 10)${\left( \dfrac{x-4}{2}\right)}$

= ${\dfrac{2x\left( x-4\right) +10\left( x-4\right) }{2}}$

= ${\dfrac{2x\times x-2x\times 4+10\times x-10\times 4}{2}}$

= ${\dfrac{2x^{2}-8x+10x-40}{2}}$

By combining the like terms, we get,

${\dfrac{2x^{2}+\left( -8x+10x\right) -40}{2}}$

Adding the like terms, we get,

${\dfrac{2x^{2}+2x-40}{2}}$

Before divisions, we must find the common factors among all the terms and represent the remaining ones.

By finding the common factors, we get,

${\dfrac{2\left( x^{2}+x-20\right) }{2}}$

Now, dividing the expressions, we get,

${x^{2}+x-20}$

Thus, 2x + 10 and ${\dfrac{x-4}{2}}$ can be simplified as ${x^{2}+x-20}$.

Sometimes, we use some algebraic expression formulas to easily solve the expressions or equations.

Formulas

Following is a list that shows some of the general formulas.

${\left( a+b\right) ^{2}}$${a^{2}+2ab+b^{2}}$
${\left( a-b\right) ^{2}}$${a^{2}-2ab+b^{2}}$
${a^{2}-b^{2}}$(a + b)(a – b)
(x + a)(x + b)${x^{2}+x\left( a+b\right) +ab}$
${\left( a+b\right) ^{3}}$${a^{3}+3a^{2}b+3ab^{2}+b^{3}}$
${\left( a-b\right) ^{3}}$${a^{3}-3a^{2}b+3ab^{2}-b^{3}}$
${a^{3}+b^{3}}$${\left( a+b\right) \left( a^{2}-ab+b^{2}\right)}$
${a^{3}-b^{3}}$${\left( a-b\right) \left( a^{2}+ab+b^{2}\right)}$

Solved Examples

Which algebraic expression is a polynomial (check all that apply and write the type):
a) 4 – 5x + 2y
b) 2x + 5 – 2(x +2) – 1

c) ${4-7x^{2}+5+2x^{2}}$
d) ${5x^{3}+2x^{2}-8x+17x^{3}-15+8x}$

Solution:

a) Here, the unlike terms are 4, – 5x, 2y.
Thus, 4 – 5x + 2y is a trinomial expression, and it is a polynomial.
b) The given expression is 2x + 5 – 2(x +2) – 1
Now, by solving the brackets, we get,
2x + 5 – 2x – 4 – 1
By combining the like terms, we get,
(2x – 2x) + (5 – 4 – 1)
= 0
Thus, 2x + 5 – 2(x +2) – 1 is not a polynomial expression.
c) The given expression is ${4-7x^{2}+5+2x^{2}}$
Now, by combining the like terms, we get,
(4 + 5) + ${\left( -7x^{2}+2x^{2}\right)}$
= 9 – ${5x^{2}}$
Here, the unlike terms are 9, ${-5x^{2}}$.
Thus, ${4-7x^{2}+5+2x^{2}}$ is a binomial expression, and it is a polynomial.
d) The given expression is ${5x^{3}+2x^{2}-8x+17x^{3}-15+8x}$
Now, by combining the like terms, we get,
${\left( 5x^{3}+17x^{3}\right) +2x^{2}+\left( 8x-8x\right) -15}$
= ${22x^{3}+2x^{2}-15}$
Here, the unlike terms are ${22x^{3}}$, ${2x^{2}}$, – 15.
Thus, ${5x^{3}+2x^{2}-8x+17x^{3}-15+8x}$ is a trinomial expression, and it is a polynomial.

Simplify: 
a) ${\dfrac{9x^{2}y}{15x^{3}}}$
b) ${6x^{2}y+4x^{2}-3x^{2}y-2x^{2}}$
c) (x + 5)(x – 4)

Solution:

a) The given expression is ${\dfrac{9x^{2}y}{15x^{3}}}$
By finding all the factors, we get,
${\dfrac{3\times 3\times x\times x\times y}{3\times 5\times x\times x\times x}}$
= ${\dfrac{3y}{5x}}$
Thus, the simplified expression is ${\dfrac{3y}{5x}}$. 
b) The given expression is ${6x^{2}y+4x^{2}-3x^{2}y-2x^{2}}$
By combining all the like terms, we get,
${\left( 6x^{2}y-3x^{2}y\right) +\left( 4x^{2}-2x^{2}\right)}$
= ${3x^{2}y+2x^{2}}$
Thus, the simplified expression is ${3x^{2}y+2x^{2}}$.
c) The given expression is (x + 5)(x – 4)
Now, multiplying the factors, we get,
x(x – 4) + 5(x – 4)
= ${x^{2}-4x+5x-20}$
By combining the like terms, we get,
${x^{2}+\left( -4x+5x\right) -20}$
= ${x^{2}+x-20}$
Thus, the simplified expression is ${x^{2}+x-20}$.

Add the following algebraic expressions:
a) 3x + 7 and x + 2y + 9
b) ${7x^{2}+3x+5}$ and ${-3x^{2}+7x+y-9}$

Solution:

a) Given, (3x + 7) + (x + 2y + 9)
= 3x +7 + x + 2y + 9
By combining all the like terms, we get,
(3x + x) + 2y + (7 + 9)
= 4x + 2y + 16
Thus, the required sum is 4x + 2y + 16.
b) Given, (${7x^{2}+3x+5}$) + (${-3x^{2}+7x+y-9}$)
= ${7x^{2}+3x+5}$ + ${-3x^{2}+7x+y-9}$
By combining all the like terms, we get,
(${7x^{2}}$ ${-3x^{2}}$) + (3x + 7x) + y + (5 – 9)
= ${4x^{2}}$ +10x + y – 4
Thus, the required sum is ${4x^{2}}$ +10x + y – 4.

Subtract: $({7x^{2}+3x+5)}$ – ${(-3x^{2}+7x+y-9)}$

Solution:

Given, (${7x^{2}+3x+5}$) – (${-3x^{2}+7x+y-9}$)
= ${7x^{2}+3x+5}$ + ${3x^{2}-7x-y+9}$
By combining all the like terms, we get,
(${7x^{2}}$ + ${3x^{2}}$) + (3x – 7x) – y + (5 + 9)
= ${10x^{2}}$ – 4x – y + 14
Thus, the required sum is ${10x^{2}}$ – 4x – y + 14.

Last modified on January 15th, 2024

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