Polynomial Division: Definition, Types, Theorem

Division of polynomials concept is very similar to the normal division. As a polynomial contains a long string of monomials, each of the monomials is taken into consideration while dividing it. The long division method is the most commonly used division method for polynomials. The number of monomials (terms) in a polynomial of the dividend and divisor might differ but they all follow a similar division method.

Key Takeaways - Monomial, Binomial, Polynomial, Long Division Method, Degree of Polynomial, Remainder theorem, Roots of Polynomial

Read Also: Multiplying a Monomial by a Polynomial

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Order and Division of Polynomials

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Polynomial is an algebraic expression of the type 

a_nxⁿ + a_n−1xⁿ⁻¹+…………………a₂x² + a₁x + a₀

where “n” is either 0 or positive variables and real coefficients.

It has more than two terms. It is important to arrange the terms of a polynomial according to their ‘order’ before division. The higher-order terms are placed first followed by other terms ascendingly. For eg. in the polynomial x² + 2x – 7, the term x² has the highest order of ‘2’ followed by 2x with the order ‘1’ and 7 with the order ‘0’. 

Terms of Polynomial

The order of the term in a polynomial is decided based on the sum of the highest power to which x (unknown variable) is raised. There are many methods of division for polynomials, but the long division method is applicable for all types of polynomials and is simpler to understand. Concisely, it deals with the division of every term of the divisor with every term of the dividend. Based on the number of terms, an expression is named as monomial, binomial, and polynomial.

The video below explains this:

Polynomials Detailed Video Explanation:

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Types of Polynomial Division

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The different types for polynomial division are given below:

Division of a monomial with a monomial

A monomial is an expression which has only one term. It has the simplest division. Let us consider 60x2 divided by 10x, 

\(\frac{60x^2}{10x} = \frac{6×10×x×x}{1×10×x} = 6x\)

Division of a polynomial with a monomial

A polynomial contains more than two terms. Each term of the polynomial is treated separately and divided individually with the monomial. 

For eg. Divide 24x³ – 12xy + 9x by 3x

Firstly, we need to check if the polynomial is arranged in the ascending order of its terms. 24x³ has the order 3, -12xy has the order 2 (adding both the exponents of the unknown terms x₁ and y₁), 9x has the order 1. Thus, the polynomial is properly arranged. Here the dividend is the polynomial 24x³ – 12xy + 9x and the divisor is the monomial 3x. Now using the long division method, 

\(\frac{24x^3-12xy+9x}{3x}\)

Now divide every term of the dividend with the monomial 3x,

8x² – 4y +3

We need to find such a term for the quotient which when multiplied with the divisor will give the dividend’s each term as the answer. The answer is the quotient 8x2 – 4y +3.

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Division of a Polynomial with a Binomial

It is important that both the binomial and the polynomial are arranged in the order of their terms before division. A binomial having two terms, each of the terms is divided individually with the polynomial. The binomial is the divisor, and the polynomial is the dividend. 

For eg. Divide 3x³ – 8x + 5 with x – 1

The polynomial 3x³ – 8x + 5 has 3,1,0 as the order of its terms. There is a missing order which can be written as 0x² to complete the ascending order of the polynomial. Thus the polynomial is 3x³ + 0x² – 8x + 5. The binomial x - 1 has the order 1,0.

3x²

Here, the first term of the binomial ‘x’ is used to divide the first term of the polynomial. The term 3x² is written in the quotient because we know that by multiplying the first term of the binomial ‘x’ with ‘3x2’ will give us the first term of the polynomial ‘3x³’. After getting the quotient ‘3x²’ the second term of the binomial ‘-1’ is multiplied with the same quotient ‘3x²’ and its answer ‘-3x²’ is written next to the previous answer. By subtracting, all the signs of the terms change, and the remaining terms of the polynomials are carried forward for further division.

3x² + 3x - 5

The division continues further with an aim to eliminate all the terms by adding apt terms to the quotient. 

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Division of a polynomial with a polynomial

This type of division proceeds very similar to the previous type, the only difference is that the divisor will now have more than two terms in its expression.

For eg. Divide x² + 2x + 3x³ + 5 by 1 + 2x + x²

Arranging the dividend x² + 2x + 3x³ + 5 in its order → 3x³ + x² + 2x + 5

Arranging the divisor 1 + 2x + x² in its order » x² + 2x + 1

Obtain the first term of the quotient in such a way that multiplying it with the first term of the divisor will give the first term of the dividend as the answer in order to eliminate it through subtraction. Continue to multiply the same term of the quotient with the remaining terms of the divisor and fill in the answers below the terms of the dividend of a similar order. 

3x - 5

The division cannot continue further as the degree of the remainder is less than the degree of the divisor.

Read Also: Permutations and Combinations 

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Remainder Theorem

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While dividing 15 by 6, we get 2 as the quotient and 3 as the remainder. So, in an equation form it is expressed as 15 = (6 × 2) + 3 as per the formula –

Dividend = (Divisor × Quotient) + Remainder

The Remainder theorem in polynomials helps in finding the remainder of a polynomial without actually dividing it. Let us consider p(x) as the dividend, x – a as the divisor, q(x) as quotient, and r(x) as remainder. Using the above given formula,

p(x) = (x - a) × q(x) + r(x)

The remainder is always constant so we can also write as,

p(x) = (x - a) × q(x) + r

Substituting x as a,

p(a) = (a - a) × q(a) + r

p(a) = (0) × q(a) + r

p(a) = r

Hence it is proved that we can find the remainder of any division if we know the ‘a’ term of the divisor.

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Things to Remember 

• Ensure to set the terms of a polynomial in standard order before proceeding to divide it.
• If there is a missing order in the series of terms in a polynomial, one should add 0xn, where n is the value of the missing order.
• The degree of a polynomial is the value of the highest order of its terms.
• Long division method concisely deals with the division of every term of the divisor with every term of the dividend.

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