Division of Algebraic Expressions: Definition, Terms, Types

Algebraic expressions are the combination of variables and constants connected by the four fundamental arithmetic operations such as addition, subtraction, multiplication, and division. The division of algebraic expressions is the inverse process of the multiplication of algebraic expressions. You can divide the algebraic expressions by utilising the division of a monomial by another monomial, division of a polynomial by a polynomial, and division of algebraic expressions using the long division method.

In addition and subtraction, we add or subtract coefficients, respectively, and the variables remain the same. Similarly, on multiplying any two algebraic expressions, their constants are multiplied with the constants, and the variables are multiplied with the variables. This article details algebraic expression, types, and how to carry out the division of algebraic expressions.

Basic Terms Related to Algebraic Expressions

Some basic terms used in algebraic expressions are:

Constant: A symbol having a fixed numerical value is called a constant.

For example: \(5,50,500,0,-10\), etc.

Variable: A variable is a symbol that takes various numerical values.

For example: In \(3x+5, 5\) is the constant term, and \(x\) is the variable.

Terms: Various parts of an algebraic expression, separated by \(‘+’\) or \(‘-‘\), are called terms of the expression.

For example, \(3x+2y\) is an algebraic expression consisting of three terms

Factors: If an algebraic expression can be written as the product of algebraic expressions, then each of these expressions is called the factors of the given algebraic expression.

For example: In the algebraic expression \(4 x^{2}+2 x, 2 x\) and \((2 x+1)\) are the factors.

Coefficients: In a term of an algebraic expression, the numeral factor of a term is called a coefficient, and any of the factors with the sign of the term is called the coefficient of the product of the other factors.

For example: In \(3 x y\), the coefficient of \(y\) is \(3 x\), the coefficient of \(x\) is \(3 y\), and the coefficient of \(x y\) is \(3\).

Like and Unlike Terms: The terms having the same algebraic factors are known as like terms; otherwise, they are called, unlike terms.

For example: In \(2 a^{2} b-7 a b-4 b a^{2}, 2 a^{2} b\) and \(-4 b a^{2}\) are like terms, and \(-7 a b\) is unlike term.

Algebraic Expressions

Algebraic expressions are the combination of variables and constants connected by the four fundamental arithmetic operations such as addition, subtraction, multiplication, and division. An algebraic expression looks like this:

Types of Algebraic Expressions

Based on the number of terms, we can classify algebraic expressions into three types, monomial, binomial, and trinomial.

Monomial: An algebraic expression having only one term is called a monomial.

Examples of monomials are \(5,2 x, 3 a^{2}, 4 x y\), etc.

Binomial: An algebraic expression having two terms separated by either the addition \((+)\) or subtraction sign \((-)\) is called a binomial.

Examples of binomial expressions are \(2 x+3,3 x-1,2 x+5 y, 6 x-3 y\), etc.

Trinomial: An algebraic expression having exactly three terms is called trinomial.

Examples of trinomials are \(4 x^{2}+9 x+7,12 p q+4 x^{2}-10,3 x+5 x^{2}-6 x^{3}\) etc.

Quadrinomial: An algebraic expression having exactly four terms is called a quadrinomial.

Examples of quadrinomials are \(4 x^{2}+3 x^{3}+9 x+7,12 p q+4 s+4 x^{2}-10,3 x^{4}+3 x+5 x^{2}-6 x^{3}\) etc.

Polynomial: Algebraic expressions having one or more terms, with the power of variables as whole numbers are called polynomials.

Division of Algebraic Expressions

In algebra, we can perform the division of algebraic expressions in three ways:

1. Division of a monomial by another monomial
2. Division of a polynomial by a monomial
3. Division of a polynomial by another polynomial

The basic division algorithm formula for dividing numbers is something that we are all familiar with.
\({\text{ Dividend }} = ({\text{ Divisor }} \times {\text{ Quotient }}) + {\text{ Remainder}}\)

Division of a Monomial by Another Monomial

When dividing a monomial by another monomial, there are two rules to keep in mind:

Rule 1: The quotient of two monomials’ coefficients is equal to the quotient of their coefficients.

Rule 2: In a quotient of two monomials, the variable portion equals the quotient of the variables in the given monomials.

Example: Divide \(12 x^{3} y^{2}\) by \(3 x^{2} y\)
\(\frac{12 x^{3} y^{2}}{3 x^{2} y}=\frac{12 \times x \times x \times x \times y \times y}{3 \times x \times x \times y}=4 \times x \times y=4 x y\)

Division of a Polynomial by a Monomial

In this case, we can use two methods:

1. We are dividing each term of the given polynomials by the monomial. This method is called the cancellation method.
2. We can take out the common factors in the polynomial. If these factors are similar to monomial, they can be cancelled by the monomial factor. This method is called the common factor method.

Example: Divide \(x^{3} y^{3}+x^{2} y^{3}-x y^{4}+x y\) by \(x y\) using the cancellation method and common factor method.

Method 1: Cancellation method
We will divide each term of the polynomial by given monomial \(\frac{x^{3} y^{3}+x^{2} y^{3}-x y^{4}+x y}{x y}=\frac{x^{3} y^{3}}{x y}+\frac{x^{2} y^{3}}{x y}-\frac{x y^{4}}{x y}+\frac{x y}{x y}\)

\(=x^{2} y^{2}+x y^{2}-y^{3}+1\)

Method 2: Common factor method We will express each term of polynomial in factor form; then, we will separate common factors.

\(x^{3} y^{3}+x^{2} y^{3}-x y^{4}+x y\)
\(=x \times x \times x \times y \times y \times y+x \times x \times y \times y \times y-x \times y \times y \times y \times y+x \times y\)
\(=x y(x \times x \times y \times y+x \times y \times y-y \times y \times y+1)\)

Then, \(x^{3} y^{3}+x^{2} y^{3}-x y^{4}+x y \div x y=\frac{x y(x \times x \times y \times y+x \times y \times y-y \times y \times y+1)}{x y}\)
\(=x^{2} \times y^{2}+x \times y^{2}-y^{3}+1=x^{2} y^{2}+x y^{2}-y^{3}+1\)

Division of a Polynomial by a Polynomial

We must divide the algebraic expressions in the numerator and denominator into irreducible factors and cancel the common factors when dividing a polynomial by another polynomial.

Example: Divide \(y^{2}+18 y+65\) by \(y+5\)
Solution: We have to factorize \(y^{2}+18 y+65\).
Since \(13\) and \(5\) are two integers such that \(13 \times 5=65\) and \(13+5=18\)
On putting these values in the given expression, we get \(y^{2}+18 y+65=y^{2}+(13+5) y+13 \times 5\)
\(=y^{2}+13 y+5 y+13 \times 5\)
\(=y(y+13)+5(y+13)=(y+13)(y+5)\) (by taking out the common factor \((y+5)\))
\(\therefore \frac{y^{2}+18 y+65}{y+5}=\frac{(y+13)(y+5)}{y+5}=y+3\)

Division of Algebraic Expressions by Long Division Method

A method of division of a polynomial by another polynomial of the same or lower degree is known as the long division of algebraic expressions.

In the polynomial long division method, the numerator and denominator are both polynomials, as given below.

The long division of polynomials consists of a divisor, a quotient, a dividend, and a remainder.

Steps for Long Division of Algebraic Expressions

The method for dividing by a polynomial with more than one term (the long division method) are as follows:

Step 1: Arrange the polynomial from the greatest degree to the smallest degree (descending order). If any term is missing, then write \(0\) in that place.
For example: By using the long division method, let us divide the polynomial \(a(x)=6 x^{4}+3 x-9 x^{2}+6\) by the quadratic polynomial \(b(x)=x^{2}-2\).

First, arrange the given polynomial in the descending order of its degree.

Step 2: Write \(0\) as the coefficient in the missing terms. Divide the polynomial \(a(x)\) by \(b(x)\) using the same method to divide the numbers. \(a(x): 6 x^{4}+0 x^{3}-9 x^{2}+3 x+6\)

\(b(x): x^{2}+0 x-2\)

Step 3: Then divide the first term of the dividend by the first term of the divisor. Now, divide \(6 x^{4}\) by \(x^{2}\) we get the \({1^{{\rm{st}}}}\) term of the quotient is \(6 x^{2}\).

Step 4: Multiply the quotient obtained in the previous step by the divisor. That is, multiply the divisor by \(6 x^{2}\).
\(6 x^{2} \times\left(x^{2}+0 x-2\right)\)
Then, we will get the product under the dividend as
\(6 x^{4}+0 x^{3}-12 x^{2}\)

Step 5: To get the new dividend, subtract the obtained product from the dividend and then put down the successive terms.

Step 5: We should repeat steps \(2,3\), and \(4\) until there are no more terms to bring down. Divide \(3 x^{2}\) by \(x^{2}\), we get the \({2^{{\rm{nd}}}}\) term of the quotient equals \(3\).

The power of the remaining dividend \(3 x\) is \(1\). It is less than the power of the divisor, which is \(2\). As a result, we get the required non-zero remainder.

Note: Since the remainder is non-zero, so we can say that \(x^{2}-2\) is not a factor of \(6 x^{4}-9 x^{2}+3 x+6\). If the remainder is zero, then we can say that \(x^{2}-2\) is a factor of \(6 x^{4}-9 x^{2}+3 x+6\).

Learn Important Algebra Formulas

Solved Examples on Division of Algebraic Expressions

Q.1. Divide \(6 x y-4 y+6-9 x\) by \((2 y-3)\)
Ans: Consider \(6 x y-4 y+6-9 x\)
Taking \(2 y\) as common in the first two terms and \(3\) as common in the successive two terms, we get
\(2 y(3 x-2)+3(2-3 x) \Rightarrow 2 y(3 x-2)-3(3 x-2)\)
\(=(2 y-3)(3 x-2)\)
Therefore, factors of \(6 x y-4 y+6-9 x\) are \((2 y-3)(3 x-2)\).
Now, \(\frac{6 x y-4 y+6-9 x}{(2 y-3)}=\frac{(2 y-3)(3 x-2)}{(2 y-3)}=(3 x-2)\)

Q.2. Find the remainder when we divide \(x^{4}+x^{3}-2 x^{2}+x+1\) by \(x-1\).
Ans: It is given that dividend \(=x^{4}+x^{3}-2 x^{2}+x+1\) and divisor \(=x-1\).
Let us divide the polynomial.

So, here the quotient is \(x^{3}+2 x^{2}+1\), and the remainder is \(2\).
Hence, \(2\) is the remainder when \(x^{4}+x^{3}-2 x^{2}+x+1\) is divided by \(x-1\).

Q.3. Divide \(16 x^{4} y^{2}\) by \(4 x y^{2}\).
Ans: We need to divide \(16 x^{4} y^{2}\) by \(4 x y^{2} . \frac{16 x^{4} y^{2}}{4 x y^{2}}=\frac{16 \times x \times x \times x \times x \times y \times y}{4 \times x \times y \times y}=4 \times x \times x \times x=4 x^{3}\)

Q.4. By using the long division method, divide the polynomial \(3 x^{2}-x^{3}-3 x+5\) by the trinomial \(x-1-x^{2}\) and verify the division algorithm.
Ans: Dividend \(=-x^{3}+3 x^{2}-3 x+5\) and divisor \(=-x^{2}+x-1\).

Since degree \((3)=0<2\), the degree \(\left(-x^{2}+x-1\right)\), so, quotient \(=x-2\), remainder \(=3\)
Now, the division algorithm is
\({\text{Divisor}} \times {\text{Quotient}} + {\text{Remainder}}\)
\(=\left(-x^{2}+x-1\right)(x-2)+3\)
\(=-x^{3}+x^{2}-x+2 x^{2}-2 x+2+3\)
\(=-x^{3}+3 x^{2}-3 x+5=\) Dividend
Hence, the division algorithm is verified.

Q.5. Divide the polynomial \(12-14 a^{2}-13 a\) by the binomial \(3+2 a\) and find the quotient and the remainder.
Ans: Dividend \(=-14 a^{2}-13 a+12\) and divisor \(=2 a+3\).

Quotient \(=-7 a+4\), Remainder \(=0\).

Summary

Algebraic expressions are the combination of variables and constants through the four fundamental arithmetic operations like addition, subtraction, multiplication, and division. Some of the common terms of algebraic expressions are variables, constants, factors, coefficients, etc.

The division of algebraic expressions is referred to as the inverse process of the multiplication of algebraic expressions. While dividing the algebraic expressions, we cancel the common terms, which is similar to the division of the numbers. Furthermore, the division of algebraic expressions generally includes at least 1 operational symbol and 1 variable.

Learn About Factorisation of Algebraic Expressions

FAQs on Division of Algebraic Expressions

Q.1. How do you divide algebraic expressions?
Ans: Using the cancellation method, common factors method, or using the long division method, we can easily divide the algebraic expressions to any degree.

Q.2. What is an example of a division expression?
Ans: Suppose we want to write the division expression for “one-fourth of \(5\) times the difference between a number and \(8\)”.
Let \(x\) be the number.
Then, we have \(\frac{5(x-8)}{4}\).

Q.3. Can you divide algebraic expressions?
Ans: Yes, we can divide the algebraic expressions by using the division of a monomial by another monomial, division of a polynomial by a polynomial, and division of algebraic expressions using the long division method.

Q.4. What do you mean by like and unlike terms?
Ans: The terms having the same algebraic factors are known as like terms; otherwise, they are called, unlike terms.
For example: In \(2 a^{2} b-7 a b-4 b a^{2}, 2 a^{2} b\) and \(-4 b a^{2}\) are like terms, and \(-7 a b\) is unlike term.

Q.5. Can the division of an algebraic expression present without an operational symbol?
Ans: No, the division of algebraic expressions should include at least \(1\) operational symbol and \(1\) variable like \(\frac{x}{2}\).

Now you are provided with all the necessary information regarding the division of algebraic expressions. Practice more questions and master this concept. Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.

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