Division of Polynomials: Polynomials are algebraic expressions consisting of variables and constants such that the exponent on the variables is a whole number. We can perform arithmetic operations such as addition, subtraction, multiplication, and division with polynomials. While using division operation we divide polynomial from a polynomial. It is to be noted that the division of 2 polynomial may or may not result in a polynomial.

Polynomial is derived from the Greek word. Poly means many and nomial means terms, so together, we can call a polynomial as many terms. So a polynomial has one or more than one term. The division of polynomials follows the same rules that we use to follow in the division of integers. This article details polynomials, their degree, types, and how to carry out the division of polynomials.

Polynomial Definition

Polynomials are algebraic expressions consisting of variables and constants with a whole number exponents of the variables. A polynomial looks like this:

A polynomial \(p(x)\) in one variable \(x\) is an algebraic expression in \(x\) of the form
\(p(x) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} +  \ldots  + {a_2}{x^2} + {a_1}x + {a_0}\)
where \({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are constants and \({a_n} \ne 0\).
\({a_0},\,{a_1},\,{a_2},\, \ldots ,\,{a_n}\) are respectively the coefficients of \({x^0},{x^1},{x^2},…,{x^n}\), and \(n\) is called the degree of the polynomial, which should be a whole number.

Each of \({a_n}{x^n},\,{a_{n – 1}}{x^{n – 1}},\, \ldots ,\,{a_0}\), with \({a_n} \ne 0\), is called a term of the polynomial \(p(x)\). It is also important to note that a polynomial can’t have fractional or negative exponents.
Examples of polynomials are \(3{y^2} + 2x + 5,\,{x^3} + 2{x^2} – 9x – 4,\,10{x^3} + 5x + y,\,4{x^2} – 5x + 7\) etc.

Types of Polynomials

Polynomials are classified based on the number of terms and their degree. Let’s see both types of classification.

Based on the Number of Terms

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

1. Monomial: A polynomial having only one term is called a monomial.
Examples of monomials are \(5,\,2x,\,3{a^2},\,4xy\), etc.

2. Binomial: A polynomial having two terms separated by either the addition \(( + )\) or subtraction sign \(( – )\) is called a binomial.
Examples of binomial expressions are \(2x + 3,\,3x – 1,\,2x + 5y,\,6x – 3y\), etc.

3. Trinomial: A polynomial having exactly three terms is called trinomial.
Examples of trinomials are \(4{x^2} + 9x + 7,\,12pq + 4{x^2} – 10,\,3x + 5{x^2} – 6{x^3}\) etc.

Based on the Degree of a Polynomial

Based on the degree, we can classify the polynomials as zero or constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic or fourth-degree polynomial, and so on.

1. Constant or Zero Polynomial: A polynomial whose power of the variable is zero is known as a constant or zero polynomial. When the power of the variable is zero, its value is nothing but \(1\) as \({x^0} = 1\). The zero polynomials will have terms that are constants like \(2,\,5,\,10,\,101\) etc.
Example: \(3{x^0} = 3 \times 1 = 3\).

2. Linear Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(1\) is a linear polynomial.
Example: \(x – 1,\,y + 1,\,a + 4\), etc.

3. Quadratic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(2\) is a quadratic polynomial.
Example: \({x^2} + x,\,{y^2} + 1,\,{a^2} + 8\), etc.

4. Cubic Polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(3\) is a cubic polynomial.
Example: \({y^3} + 8,\,{x^3} – 27,\,5 + {a^3},\,{x^3} + {x^2} – x + 2\) etc.

5. Quartic Polynomial or fourth-degree polynomial: A polynomial whose highest power of the variable or the polynomial degree is \(4\) is known as a quartic polynomial or fourth-degree polynomial.
Example: \({x^4} + {x^3} – {x^2} + x + 1,\,{y^4} – {y^2} + 1\), etc.

Division of Polynomials Methods

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

We all are familiar with the basic division algorithm formula of dividing numbers.
\({\rm{Dividend = (Divisor \times Quotient) + Remainder}}\)

A long division of polynomials is a method for dividing a polynomial by another polynomial of the same or a lower degree. In the long division of polynomials, numerator and denominator are polynomials, as shown below.

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

Steps for Long Division of Polynomials

The following are the steps required for dividing by a polynomial containing more than one term (Long division method):

Step 1: Write the polynomial in descending order(arranged from the largest degree to the smallest degree). If any term is missing, then use a zero in place of the missing term.
For example: Let us divide the polynomial \(a(x) = 6{x^4} + 3x – 9{x^2} + 6\) by the quadratic polynomial \(b(x) = {x^2} – 2\) by using the long division method.

First, arrange the given polynomial in the descending order of the power of the variable.

Step 2: Add the missing terms with zero as the coefficient. Divide the polynomial \(a(x)\) by \(b(x)\) using the same method that we use to divide the numbers.
\(a(x):6{x^4} + 0{x^3} – 9{x^2} + 3x + 6 = {x^2} – 2\)
\(b(x):{x^2} + 0x – 2\)

Step 3: 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 \({{\rm{1}}^{{\rm{st}}}}\) 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 \(6{x^2}\).
\(6{x^2} \times \left( {{x^2} + 0x – 2} \right)\)
Then, we will get the product under the dividend as
\(6{x^4} + 0{x^3} – 12{x^2}\)

Step 5: Subtract the obtained product from the dividend and then write down the next terms to get the new dividend.

Step 5: Repeat the process in Steps \(2,\,3\), and \(4\) until we get remainder or no more terms to bring down.
Divide \(3{x^2}\) by \({x^2}\), we get the \({{\rm{2}}^{{\rm{nd}}}}\) term of the quotient equals \(3\).

The power of the remaining dividend \(3x\) is \(1\). It is less than the power of the divisor which is \(2\). So, 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 \(6{x^4} – 9{x^2} + 3x + 6\).
If the remainder is zero then we can say that \({x^2} – 2\) is a factor of \(6{x^4} – 9{x^2} + 3x + 6\).

Solved Examples

Q.1. Divide the polynomial \(2{x^2} + 3x + 1\) by \(x + 2\).
Ans:

So, here the quotient is \(2x – 1\) and the remainder is \(3\).
Also, \((2x – 1)(x + 2) + 3 = 2{x^2} + 3x – 2 + 3 = 2{x^2} + 3x + 1\)
i.e., \(2{x^2} + 3x + 1 = (x + 2)(2x – 1) + 3\)

Therefore, \({\rm{Dividend = Divisor \times Quotient + Remainder}}\)

Q.2. Find the remainder obtained when \({x^4} + {x^3} – 2{x^2} + x + 1\) is divided by \(x – 1\).
Answer: 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. Find all the zeroes of \(2{x^4} – 3{x^3} – 3{x^2} + 6x – 2\), if the two of its zeroes are \(\sqrt 2 \) and \( – \sqrt 2 \).
Ans: Since two zeroes are \(\sqrt 2 \) and \(\sqrt 2 ,\,(x – \sqrt 2 )(x + \sqrt 2 ) = {x^2} – 2\) is a factor of the given polynomial. Now, we divide the given polynomial by \({x^2} – 2\).

So, \(2{x^4} – 3{x^3} – 3{x^2} + 6x – 2 = \left( {{x^2} – 2} \right)\left( {2{x^2} – 3x + 1} \right)\) Now, by splitting \( – 3x\), we factorise \(2{x^2} – 3x + 1\) as \((2x – 1)(x – 1)\).
So, its zeroes are \(x = \frac{1}{2}\) and \(x – 1\).

Hence, the required zeroes of the given polynomial are \(\sqrt 2 ,\, – \sqrt 2 ,\,\frac{1}{2}\), and \(1\).

Q.4. Divide the polynomial \(3{x^2} – {x^3} – 3x + 5\) by the trinomial \(x – 1 – {x^2}\), and verify the division algorithm.
Ans: Note that the given polynomials are not in standard form. To carry out division, we first write both the dividend and divisor in decreasing orders of their degrees.
So, dividend \( =  – {x^3} + 3{x^2} – 3x + 5\) and divisor \( =  – {x^2} + x – 1\).

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

Question 5: Divide the polynomial \(12 – 14{a^2} – 13a\) by the binomial \(3 + 2a\).
Answer: Note that the given polynomials are not in standard form. To carry out division, we first write both the dividend and divisor in decreasing orders of their degrees.
So, dividend \( =  – 14{a^2} – 13a + 12\)  and divisor \( = 2a + 3\).

Summary

In this article, we have learnt about polynomials, how it looks like, types of polynomials based on the number of terms like monomials, binomials and trinomials. And based on the degree, polynomials are further classified into zero-degree polynomial or constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial, quartic polynomial, etc.

Then we have discussed the method of division of polynomials by using the long division method, steps for division of polynomials, and solved examples.

FAQs – Division of Polynomials

Q.1. How do you divide polynomials with two variables?
Ans: We can divide the polynomials with two variables using the long division method or the synthetic division method.

Q.2. What is the easiest way to divide polynomials?
Ans: By using the long division method, we can easily divide the polynomials to any degree.

Q.3. How is a polynomial division used in real life?
Ans: We use polynomial division for several aspects of our daily lives. We use it in coding, engineering, business, designing, architecting, and other real-life areas. It is used to solve the problems related to solving the expressions involved in area and volume.

Q.4. How do you divide polynomials?
Ans: Polynomials can be divided the same way as we divide numbers, either by factorising or by long division. The method you use depends upon how complex are the polynomial dividend and divisor.

Q.5. How do you know if a polynomial completely divides another polynomial?
Ans: If the remainder is zero after completing the division process, then it means that the polynomials are completely divisible. If a remainder is non-zero, then the polynomial is not divisible.

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