Sample Problems on Dividing Polynomials

Problem 1. Using synthetic division, find the quotient and remainder of [Tex] \bold{\frac{x^2 + 3}{x – 4}}  [/Tex]

Solution:

Dividend = x² + 3 or, x² + 0x + 3

Divisor = x – 4

Applying synthetic division, we have:

[Tex]\begin{array}{c|rrr}&1&0&3\\4&&4&16\\\hline\\&1&4&19\\\end{array} [/Tex]

The first two numbers of the last row represent the coefficients of the quotient and the third value is the remainder.

Thus, the quotient is x + 4 and the remainder is 19.

Problem 2. Solve [Tex]\bold{\frac{4x^3+5x^2+5x+8}{4x+1}} [/Tex]using long division.

Solution:

Dividend = 4x³ + 5x² + 5x + 8

Divisor = 4x + 1

Using long division method, we have:

[Tex]\begin{array}{r} x^2+x+1\phantom{)}   \\ 4x+1{\overline{\smash{\big)}\,4x^3+5x^2+5x+8\phantom{)}}}\\ \underline{4x^3~\phantom{}+x^2~~~~~~~~~~~\phantom{-b)}}\\ 4x^2+5x~~~~~~~\phantom{)}\\ \underline{~\phantom{()}4x^2+1x~~~~~~~~~}\\ 4x+8\phantom{)}\\ \underline{-~\phantom{()}(4x+1)}\\ 7\phantom{)}\\ \end{array} [/Tex]

Thus, the quotient and remainder are x² + x + 1 and 7 respectively.

Problem 3. Solve [Tex]\bold{\frac{4x^3-3x^2+3x-1}{x-1}} [/Tex] using synthetic division.

Solution:

Dividend = 4x³ – 3x² + 3x – 1

Divisor = x – 1

Applying synthetic division, we have:

[Tex]\begin{array}{c|rrr}&4&-3&3&-1\\1&&4&1&4\\\hline\\&4&1&4&3\\\end{array} [/Tex]

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 4x² + x + 4 and the remainder is 3.

Problem 4. Solve [Tex]\bold{\left( 5{{x}^{3}}-6{{x}^{2}}+3x+11 \right)\div \left( x-2 \right)} [/Tex] using synthetic division.

Solution:

Dividend = 5x³ – 6x² + 3x + 11

Divisor = x – 2

Applying synthetic division, we have:

[Tex]\begin{array}{c|rrr}&5&-6&3&11\\2&&10&8&22\\\hline\\&5&4&11&33\\\end{array} [/Tex]

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 5x² + 4x + 11 and the remainder is 33.

Problem 5. Solve [Tex]\bold{\left( 18{{x}^{4}}+9{{x}^{3}}+3{{x}^{2}} \right)\div \left( 3{{x}^{2}}+1 \right) } [/Tex] using long division.

Solution:

Dividend = 18x⁴ + 9x³ + 3x² + 0x + 0

Divisor = 3x² + 1

Using long division method, we have:

[Tex]\begin{array}{r} 6x^2+3x-1\phantom{)}   \\ 3x^2+1{\overline{\smash{\big)}\,18x^4+9x^3+3x^2+0x+0\phantom{)}}}\\ \underline{18x^4~\phantom{}+0x^3+6x^2~~~~~~~~~\phantom{-b)}}\\ 9x^3-3x^2+0x+0\phantom{)}\\ \underline{~\phantom{()}9x^3+0x^2+3x~~~~~~~~~}\\ -3x^2-3x+0\phantom{)}\\ \underline{~\phantom{()}-3x^2+0x-1}\\ -3x+1\phantom{)}\\ \end{array} [/Tex]

Thus, the quotient and remainder are 6x² + 3x – 1 and -3x + 1 respectively.

Dividing Polynomials as the name suggests the process of division of a polynomial by another polynomial. It is also called Polynomial Division. As we know, any expression with one variable with its various powers and coefficients is called a polynomial. The most general form of a polynomial is given as:

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

Where a₀, a₁, a₂, . . ., a_n are the real coefficients. In Dividing Polynomial we divide the polynomial with a higher degree by a polynomial (that can monomial, binomial, trinomial, or any other higher degree polynomial) with less degree.

In this article, we will learn about all the necessary topics for the division of polynomials such as methods of division i.e., Long Division Method, Synthetic Division, etc. Also, we will learn how to solve problems related to Dividing Polynomials.