## What is an Algebraic Expression?

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Example: $7x-4y+2$ , $8z-3$.

Note: Algebraic expression and algebraic equation are two different terms. The algebraic equation consists of two or more algebraic expressions which are separated by an equal sign$(=)$.

## Parts of an Algebraic Expression

### Constant

Any term that does not change throughout is called a constant.

### Variable

Any symbol, alphabet, or any letter that is not known is called a variable.

### Coefficient

The coefficient is a multiplied factor with any variable in an expression that modifies it. It may be constant.

### Mathematical Operator

The symbols which tell about the mathematical operations that need to be performed are referred to as mathematical operators.

Now, let us understand the above terms with an example.

Example: $7x-4y+2$.

Here,

- $x$ and $y$ are the variables
- $7$ and $-4$ are the coefficients of $x$ and $y$ respectively.
- $2$ is a constant.
- -, + and multiplication are the mathematical operators.

## Terms in an Algebraic Expression

Here, we will discuss the term and variable and how they are related. In mathematics, we generally come across with alphabetical letters like$x,y,z$ , etc. along with mathematical numbers. The letters are used to represent the unknown numbers so assume any random variable in place of that term. Thus, it is defined as a symbol to represent a number whose value is not known to us.

And when it is associated with any number then it is said to be a term. Like when the variable is in a multiplication or division with the number it is said to be a single term only, but when it is added or subtracted from any number then it comprises two different terms. Let’s understand this with an example.

Example: $7x-4y+2$ (From above)

Here are three terms that are: $7x$, $-4y$, and 2.

The whole combination of single or multiple terms is called an expression and that is$7x-4y+2$.

##### Types of Algebraic Expressions Based on the Number of Terms:

1. Monomial: It is made up of only a single term is called a monomial.

For example:$7x$,$-4y$, $2$ , $3xyz$ , $\sqrt{3}xy$

2. Binomial: It is made up of two terms is said to be a binomial.

For example:$7x-4y$,$2-4y$,$xyz-{x}^{2}y$, $\frac{1}{x}+2y$

3. Trinomial: It is made up of three terms is known as trinomial.

For example:$7x-4y+2$, ${x}^{2}y+2+xy$

## Polynomial

It is a special case of algebraic expression. In this variables are not in the denominator, the exponent of each is the whole number and the coefficient of each variable is a real number.

### Linear Polynomial

A polynomial in which the maximum power of the variable is $1$ or unity. Then it is said to a polynomial of degree$1$. For example: $7x-4$, $-x+3$.

### Quadratic Polynomial

A polynomial in which the maximum power of the variable is$2$. Then it is said to be a polynomial of degree$2$. For example: ${x}^{2}-x+2$, ${z}^{2}-9$.

##### Note:

The degree of a polynomial is the maximum power of the variables in the polynomial.

## Types of Terms in an Algebraic Expression

It can be categorized into two types:

### Like Terms

In this, only the coefficient of the term can change means the variable must remain unchanged. This means the term including its powers must be the same otherwise they are not like terms. For example: ${x}^{3}$ and $4{x}^{3}$; ${x}^{2}yz$ and $5{x}^{2}yz$.

Let us understand this by a different example:

Suppose a person has two baskets. The first one contains $3$ erasers and the second one contains$5$erasers. Here the eraser represents the variable part and the number of erasers represents the coefficient. Since the variable is the same in both cases; hence these two items are the same or like items.

### Unlike Terms

In this, the different or same variables are raised to different powers.

Let’s understand this by a different example:

For example: ${x}^{3}$ and $4{x}^{3}y$; ${x}^{2}yz$ and $x{y}^{2}z$.

Let suppose now a person has three baskets which contain $3$ pencils, $5$ erasers, and $7$ pencils respectively. Now the basket $1$ and basket $2$are unlike or different items. But the basket $1$ and basket $3$ are like, i.e., they have the same item.

Here are few more examples of like and unlike terms-

## What are Some Algebraic Laws?

### Commutative Law

When doing addition or multiplication, the commutative law states that the terms can be reordered.

For variables $a$ and $b$:

- $a+b=b+a$
- $a\times b=b\times a$

### Associative Law

The associative law states that one can associate phrases or components in addition or multiplication.

For variables $a$, $b$, $c$:

- $(a+b)+c=a+(b+c)$
- $(a\times b)\times c=a\times (b\times c)$

Note: Only multiplication and addition are subject to the commutative and associative laws.

### Distributive Law

The distributive law is concerned with the addition and multiplication of numbers. When you multiply a sum by a value, it gets spread evenly among the parts of the sum. For variables $a,b,c$:

- $a(b+c)=a\times b+a\times c$
- $a(b-c)=a\times b-a\times c$

## How to Evaluate Algebraic Expressions?

To evaluate it:

- Replace the variables with their respective number.
- Perform the operations in the correct order.
- Write the result.

### Example

If $x=5$ and $y=7$, then what is the value of $\frac{3x+y}{2}$?

Solution:** **To evaluate the above algebra, just substitute it.

Now substituting $x=5$ and $y=7$ in the above expression.

$\Rightarrow \frac{3x+y}{2}=\frac{3(5)+7}{2}$

$\Rightarrow \frac{22}{2}$

$\Rightarrow 11$

Hence, the required answer is 11.

## Operations with Algebraic Expressions

### Addition

Can anyone add 3 pencils and 3 erasers?

The answer is no. No one can add three pencils and three erasers as they both are two different items. Similarly in the case of terms in algebraic expressions, one cannot add two or more, unlike terms. One can only add like terms.

So, to add the like terms just add their numerical value and the variable part remains the same after addition. For example, after $3$ pencils and $3$ pencils, we get $6$ pencils just as hereafter adding the numerical value the variable term (pencil) remains the same.

One can add-

Example-1: ${x}^{3}$ and $4{x}^{3}$.

Example-2: ${x}^{2}yz$ and $5{x}^{2}yz$.

### Subtraction

Can anyone subtract 3 erasers from 3 pencils?

The answer is no. One cannot subtract three erasers and three pencils as they both are two different items. Similarly in the case of terms in algebraic expression one cannot subtract two or more, unlike terms. One can subtract only like terms.

So, to subtract one like term from another just subtract their numerical value and the variable part remain the same after subtraction. For example, after subtracting $3$ pencils and $4$ pencils, and will get the only pencil just as hereafter subtracting the numerical value the term (pencil) remains the same.

Similarly, one can subtract-

Example-1: ${x}^{3}$ from$4{x}^{3}$.

Example-2: ${x}^{2}yz$ from $5{x}^{2}yz$.

Note that $2x$ is $2$ times $x$.

Q1: Add $2{m}^{2}+mn-7n$ , $2{m}^{2}+7n+5mn$ and $(-7mn+m)$.

**Solution:**

**Step-1:** First write down all the terms separately that need to add:

**Step-2: **Collect the like terms from above and make them separate with other unlike terms:

**Step-3:** Now add the like terms:

$4{m}^{2}+(-1)mn+(0)n+m$

$\Rightarrow 4{m}^{2}-mn+m$

Answer: Hence, $4{m}^{2}-mn+m$ is the required sum.

### Multiplication

In this, there is no restriction of like and unlike terms to multiply the terms with each other. To get the cross-product of the terms, just need to multiply each term of one expression with every term of another expression. And then just simplify it.

For Example:

Q2: Multiply $4x$ and $(3x+2)$.

**Solution: **To multiply the algebraic expressions, just multiply each term of one expression to each term of the other expression.

Step-1: Write down the expressions that need to be multiplied:

$4x\times (3x+2)$

Step-2: Multiply $4x$ with each term of other expressions as the first one has only one term.

$4x\times 3x+4x\times 2$

Step-3: Multiply the separated terms :

$\Rightarrow 12x\times x+8\times x$

$\Rightarrow 12{x}^{2}+8x$

Q3: Multiply $(x+y)$ and $(3x-2y+5)$.

**Solution: **To multiply the algebraic expressions, just multiply each term of one expression to each term of the other expression.

Step-1: Write down the expressions that need to be multiplied:

$(x+y)\times (3x-2y+5)$

Step-2: Multiply $(x+y)$ with each term of other expressions.

$x\times (3x-2y+5)+y\times (3x-2y+5)$

$\Rightarrow (3{x}^{2}-2xy+5x)+(3xy-2{y}^{2}+5y)$

Step-3: Now collect the like terms in above and make suitable operations to simplify:

$3{x}^{2}+(-2xy+3xy)+5x-2{y}^{2}+5y$

$\Rightarrow 3{x}^{2}+xy+5x-2{y}^{2}+5y$

Answer: Hence the final product is $3{x}^{2}-2{y}^{2}+xy+5x+5y$.

### Division of Algebraic Expression

In division, first, take out the common factors. After taking the common factor out, divide the algebraic expressions. On eliminating the common term between both expressions from the numerator and denominator, the solution is obtained by the long division. Let us consider an expression for performing the division operation.

Example: Evaluate $(8{x}^{2}+16x)\xf7(x+2)$.

**Solution:** First take out the common factors.

For the expression,$8{x}^{2}+16x$, $8x$ is the common factor.

So, considering $8x$ as the common factor, the expression $8{x}^{2}+16x$ becomes:

$(8{x}^{2}+16x)=8x(x+2)$

Let's now divide the algebraic expressions,

$(8{x}^{2}+16x)\xf7(x+2)=\frac{8x(x+2)}{(x+2)}$Next, eliminate (x+2) from the numerator and denominator,

$\Rightarrow 8x$Answer: Hence $8x$ is the correct answer.

## Formulas

1.${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}$

2. ${(a-b)}^{2}={a}^{2}-2ab+{b}^{2}$

3. ${a}^{2}-{b}^{2}=(a-b)(a+b)$

4. ${(a+b)}^{3}={a}^{3}+{b}^{3}+3ab(a+b)$

5. ${(a-b)}^{3}={a}^{3}-{b}^{3}-3ab(a-b)$

6. ${a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})$

7. ${a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})$

## Common Mistake

In general, most of the people confused and consider ${x}^{2}y$ and $x{y}^{2}$ same but they both are two different terms as in one case it is $x\times x\times y$ and in another it is$x\times y\times y$. Hence, they are unlike terms.

## Context and Applications

This topic is fundamental for mathematics for $7th$ grade students as without understanding this topic they can’t go through with the mathematics.

## Related Concepts

- Polynomials
- Factorization of polynomials
- Quadratic polynomials

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