In this chapter we will learn method to divide polynomial with monomial.
Let us first revise the basic concept first.
What is Monomial ?
The algebraic expression containing only one entity is called Monomial.
The entity can be constant, variable or both.
Examples of monomial are;
\mathtt{\Longrightarrow \ 2xy}\\\ \\ \mathtt{\Longrightarrow \ 5x^{2}}\\\ \\ \mathtt{\Longrightarrow \ 10y^{3}}
What is Polynomial?
The algebraic expression with one or more entity is called polynomial.
Examples of Polynomial;
\mathtt{\Longrightarrow \ 6x^{2} y\ +\ 5x}\\\ \\ \mathtt{\Longrightarrow \ 13x^{2} +\ 18xy\ +\ 4y}\\\ \\ \mathtt{\Longrightarrow \ x^{3} yz\ +\ 2xy\ +3yz\ +\ 5zx}
I hope the basics are cleared. Let us understand the method to divide polynomial with monomial.
How to divide polynomial with monomial
Let us take the below polynomial & monomial as example.
Polynomial ⟹ \mathtt{6x^{2} y\ +\ 18xy\ +2y}
Monomial ⟹ 2x
Follow the below steps for above division;
(i) Express division in form of fraction. Here the polynomial will be at numerator and monomial at denominator.
\mathtt{\Longrightarrow \ \frac{\mathtt{6x^{2} y\ +\ 18xy\ +2y}}{2x}}
(ii) Divide each term of polynomial separately.
\mathtt{\Longrightarrow \ \frac{6x^{2} y}{2x} +\ \frac{18xy}{2x} +\ \frac{2y}{2x}}
(iii) In each division, cancel out the common factors in numerator and denominator.
\mathtt{\Longrightarrow \ \frac{3x^{2-1} y}{\cancel{2x}} +\frac{9x^{1-1} y}{\cancel{2x}} +\frac{y}{x}}\\\ \\ \mathtt{\Longrightarrow \ 2xy\ +\ 9y\ +\ yx^{-1}}
Hence, after dividing all the terms present in polynomial, you will get the solution.
Note:
The reciprocal of any variable can be written with negative exponent.
\mathtt{i.e.\ \frac{1}{x^{2}} =\ x^{-2}}
I hope you understood the whole process. Let us now solve some related questions.
Polynomial by Monomial division examples
Example 01
Divide \mathtt{12x^{3} +\ 6x^{2} +3x\ \ \ by\ \ \ 3x}
Solution
Follow the below steps;
(a) Represent the division in fraction form.
\mathtt{\Longrightarrow \ \frac{\mathtt{12x^{3} +\ 6x^{2} +3x}}{3x}}
(b) Divide each term of polynomial with monomial separately.
\mathtt{\Longrightarrow \ \frac{\mathtt{12x^{3}}}{\mathtt{3x}} +\ \frac{\mathtt{\ 6x^{2}}}{\mathtt{3x}} +\ \frac{\mathtt{3x}}{\mathtt{3x}}}
(c) Cancel out common factors from numerator and denominator.
\mathtt{\Longrightarrow \ \frac{4x^{3-1}}{\cancel{3x}} +\frac{2x^{2-1}}{\cancel{3x}} +1}\\\ \\ \mathtt{\Longrightarrow \ 4x^{2} \ +\ 2x\ +\ 1}
Hence, \mathtt{ \ 4x^{2} \ +\ 2x\ +\ 1} is the solution.
Example 02
Divide \mathtt{14y^{8} +\ 28y^{3} +21y^{2} \ \ \ by\ \ \ 7y^{2}}
Solution
Follow the below steps;
(a) Express the division in form of fraction.
\mathtt{\Longrightarrow \ \frac{\mathtt{\ 14y^{8} +\ 28y^{3} +21y^{2}}}{\mathtt{7y^{2}}}}
(b) Divide each term of polynomial separately
\mathtt{\Longrightarrow \ \frac{14y\mathtt{^{8}}}{7y^{2}} +\ \frac{\mathtt{\ 28y^{3}}}{7y^{2}} +\ \frac{21y^{2}}{7y^{2}}}
(c) Cancel out common factors from numerator and denominator.
\mathtt{\Longrightarrow \ \frac{2y^{8-2}}{\cancel{7y^{2}}} +\frac{4y^{3-2}}{\cancel{7y^{2}}} +\ 3}\\\ \\ \mathtt{\Longrightarrow \ 2y^{6} +\ 4y\ +\ 3\ }
Hence, \mathtt{\ 2y^{6} +\ 4y\ +\ 3\ } is the solution.
Example 03
Divide \mathtt{\ 6x^{3} +\ 2xy+y^{2} \ \ \ by\ \ \ x^{2}}
Solution
Follow the below steps;
(a) Express division in fraction form.
\mathtt{\Longrightarrow \ \frac{\mathtt{\ 6x^{3} +\ 2xy+y^{2}}}{\mathtt{x^{2}}}}
(b) Separately divide each term of the polynomial
\mathtt{\Longrightarrow \ \frac{6x\mathtt{^{3}}}{x^{2}} +\ \frac{\mathtt{\ 2xy}}{x^{2}} +\ \frac{y^{2}}{x^{2}}}
(c) Cancel out common factors from numerator & denominator.
\mathtt{\Longrightarrow \ \frac{6x^{3-2}}{\cancel{x^{2}}} +\frac{2x^{1-2} y}{\cancel{x^{2}}} +\ \frac{y^{2}}{x^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 6x\ +\ 2x^{-1} y\ +\ x^{-2} y^{2}}
Hence, \mathtt{\ 6x\ +\ 2x^{-1} y\ +\ x^{-2} y^{2}} is the solution.
Example 04
\mathtt{13x^{4} y^{2} +\ x^{2} y^{2} +y^{3} \ \ \ by\ \ \ 2xy}
Solution
To divide the polynomial, follow the below steps;
(a) Express division in form of fraction.
\mathtt{\Longrightarrow \ \frac{\mathtt{\ \ 13x^{4} y^{2} +\ x^{2} y^{2} +y^{3} \ \ }}{2xy}}
(b) Separately divide each polynomial
\mathtt{\Longrightarrow \ \frac{\mathtt{13x^{4} y^{2}}}{2xy} +\ \frac{\mathtt{\ x^{2} y{^{2}}}}{2xy} +\ \frac{\mathtt{y^{3}}}{2xy}}
(c) Remove common factors from numerator and denominator.
\mathtt{\Longrightarrow \ \frac{13}{2} x^{3} y\ +\frac{x y}{2} +\ \frac{x^{-1} y^{2}}{2}}
Hence, \mathtt{\frac{13}{2} x^{3} y\ +\frac{x y}{2} +\ \frac{x^{-1} y^{2}}{2}} is the solution.
Example 05
\mathtt{8m^{3} n^{-2} +m^{-2} n^{2} +n^{3} \ \ \ by\ \ \ mn}
Solution
Follow the below steps;
(a) Express division in form of fraction
\mathtt{\Longrightarrow \ \frac{\mathtt{\ \ 8m^{3} n^{-2} +m^{-2} n^{2} +n^{3}}}{mn}}
(b) Separately divide each polynomial
\mathtt{\Longrightarrow \ \frac{8m\mathtt{^{3} n^{-2}}}{mn} +\ \frac{\mathtt{\ m^{-2} n{^{2}}}}{mn} +\ \frac{n\mathtt{^{3}}}{mn}}
(c) Remove the common factors
\mathtt{\Longrightarrow \ \frac{8m\mathtt{^{3-1} n^{-2-1}}}{\cancel{mn}} +\ \frac{\mathtt{\ m^{-2-1} n{^{2-1}}}}{\cancel{mn}} +\ \frac{m^{-1} n\mathtt{^{3-1}}}{\cancel{mn}}}\\\ \\ \mathtt{\Longrightarrow \ 8m^{2} n^{-3} \ +\ m^{-3} n\ +m^{-1} n^{2} \ }