Dividing polynomials by monomials

On this post we explain how to divide polynomials by monomials. You will find examples of divisions of polynomials by monomials and practice problems on dividing polynomials by monomials.

How to divide a polynomial by a monomial

Before seeing how to do the division of a polynomial by a monomial, let’s first remember how to divide monomials by monomials, since it is necessary to know it in order to do this type of polynomial operation.

The division of two monomials consists of dividing their coefficients and subtract the exponents of the variables that have the same base. Take a look at the following example:

 12x^5\div3x^2 = \cfrac{12x^5}{3x^2}=(12\div 3) x^{5-2} = 4x^3

Now, let’s see what the division of a polynomial by a monomial consists of:

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

division of polynomials by monomials

Thus, the division of a polynomial by a monomial can be split into several divisions of monomials by monomials.

Moreover, a division of a polynomial by a monomial can also be expressed as a fraction:

\begin{array}{l}15x^4+6x^3-9x^2\div (-3x^2) = \\[3ex]=\cfrac{15x^4+6x^3-9x^2}{-3x^2}=\\[3ex]=\cfrac{15x^4}{-3x^2}+\cfrac{6x^3}{-3x^2}+\cfrac{9x^2}{-3x^2}=\\[4ex] = 5x^2-2x-3\end{array}

Note in the last example that when we divide monomials or polynomials we must also take into account the sign rules. In fact, a very typical mistake in divisions of polynomials by monomials is to miss the sign of a term.

Examples of divisions of polynomials by monomials

So that you can fully understand how to divide polynomials by monomials, below we leave you with several examples of this type of divisions:

\begin{aligned}\color{orange}\bm{\rightarrow}\color{black}\quad\displaystyle\frac{14x^3+10x}{2x}&=\frac{14x^3}{2x}+\frac{10x}{2x}\\[2ex]&=7x^2+5\end{aligned}

\begin{aligned}\color{orange}\bm{\rightarrow}\color{black}\quad\displaystyle \frac{20y^7-3y^5+17y^4}{y^3}&=\frac{20y^7}{y^3}+\frac{-3y^5}{y^3}+\frac{17y^4}{y^3}\\[2ex]&=20y^4-3y^2+17y\end{aligned}

\begin{aligned}\color{orange}\bm{\rightarrow}\color{black}\quad\displaystyle \frac{5x^9-25x^6+30x^3}{5x^2}&=\frac{5x^9}{5x^2}+\frac{-25x^6}{5x^2}+\frac{30x^3}{5x^2}\\[2ex]&=x^7-5x^4+6x\end{aligned}

\begin{aligned}\color{orange}\bm{\rightarrow}\color{black}\quad\displaystyle\frac{-16x^5-2x^3}{-4x^2}&=\frac{-16x^5}{-4x^2}+\frac{-2x^3}{-4x^2}\\[2ex]&=4x^3+\frac{1}{2}x\end{aligned}

Note that if the term in the numerator is equal to the term in the denominator, we can cancel them out. But they don’t disappear! We must write a 1 in its place!

\begin{aligned}\color{orange}\bm{\rightarrow}\color{black}\quad\displaystyle\frac{12x^5+3x^2}{3x^2}&=\frac{12x^5}{3x^2}+\frac{\cancel{3x^2}}{\cancel{3x^2}}\\[2ex]&=4x^3+1\end{aligned}

Practice problems on dividing polynomials by monomials

So that you can practice, below you have several questions with answers of divisions of polynomials. You can try to solve them yourself and check your results with the proposed solution. Then you can ask us any questions you have in the comments section, we will be glad to help you.

Problem 1

Calculate the following division of a binomial by a monomial:

\cfrac{15x^5+9x^3}{3x^2}

To divide a polynomial by a monomial we have to solve the division of each term of the polynomial by the monomial:

\begin{aligned} \cfrac{15x^5+9x^3}{3x^2}& = \cfrac{15x^{5}}{3x^2}+ \cfrac{9x^3}{3x^2} \\[2ex] & = \bm{5x^3+3x} \end{aligned}

Remember that in the division of monomials by monomials, the coefficients are divided by each other and the exponents of the powers whose bases are equal are subtracted.

 

Problem 2

Simplify the following division of a polynomial by a monomial:

\cfrac{16x^5-4x^3-20x^2}{4x^2}

To divide a polynomial by a monomial we have to divide each term of the polynomial by the monomial:

\begin{aligned}\cfrac{16x^5-4x^3-20x^2}{4x^2}& = \cfrac{16x^5}{4x^2}+ \cfrac{-4x^3}{4x^2} + \cfrac{-20x^2}{4x^2} \\[2ex] & = \bm{4x^3-x-5} \end{aligned}

Remember that in the monomial division the coefficients are divided by each other and the exponents of the powers whose base are the same are subtracted.

 

Problem 3

Divide the polynomial 12x^{10}-30x^7-18x^6+54x^4 by the monomial -6x^3.

\cfrac{12x^{10}-30x^7-18x^6+54x^4}{-6x^3}

To divide a polynomial by a monomial we have to solve the division of each term of the polynomial by the monomial:

\begin{aligned}\cfrac{12x^{10}-30x^7-18x^6+54x^4}{-6x^3}& = \cfrac{12x^{10}}{-6x^3}+ \cfrac{-30x^{7}}{-6x^3} + \cfrac{-18x^6}{-6x^3} + \cfrac{54x^4}{-6x^3} \\[2ex] & = \bm{-2x^7+5x^4+3x^3-9x} \end{aligned}

Notice that the monomial in the denominator is negative and, therefore, the signs of all divisions change.

 

Problem 4

Find the result of the following division of a polynomial by a monomial with 2 variables:

\cfrac{16x^5y^3-8x^4y+32x^3y^2}{8x^2y}

Although the polynomial and the monomial of the division have two variables, we have to apply the same method: divide each term of the polynomial by the monomial.

\begin{aligned}\cfrac{16x^5y^3-8x^4y+32x^3y^2}{8x^2y}& = \cfrac{16x^5y^3}{8x^2y}+ \cfrac{-8x^4y}{8x^2y} + \cfrac{32x^3y^2}{8x^2y}\\[2ex] & = \bm{2x^3y^2-x^2+4xy} \end{aligned}

 

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