# Division of Polynomial by Monomial

Division of polynomial by monomial means dividing the polynomials which is written as numerator by a monomial which is written as denominator to find their quotient.

For example: 4a3 – 10a2 + 5a ÷ 2a

Now the polynomials (4a3 – 10a2 + 5a) is written as numerator and the monomial (2a) is written as denominator.

Therefore, we get $$\frac{4a^{3} - 10a^{2} + 5a}{2a}$$

Now we observe that there are three terms in the polynomial so, each term of the polynomial (numerator) is separately divided by the same monomial (denominator).

$$\frac{4a^{3}}{2a} - \frac{10a^{2}}{2a} + \frac{5a}{2a}$$

Note:

The process is exactly converse of finding the L.C.M. of fractions and reducing the expression into a single fraction.

Now we will cancel out the common factor from both numerator and denominator to simplify.

$$4a^{2} - 5a + \frac{5}{2}$$

Solve examples on division of polynomial by monomial:

1. Divide x6 + 7x5 – 5x4 by x2

= x6 + 7x5 – 5x4 ÷ x2

= $$\frac{x^{6} + 7x^{5} - 5x^{4}}{x^{2}}$$

Now, we need to divide each term of the polynomial by the monomial and then simplify.

= $$\frac{x^{6}}{x^{2}} + \frac{7x^{5}}{x^{2}} - \frac{5x^{4}}{x^{2}}$$

Now each term will be simplified by cancelling out the common factor.

= $$x^{4} + 7x^{3} - 5x^{2}$$

2. Divide a2 + ab – ac by –a

= a2 + ab – ac ÷ -a

= $$\frac{a^{2} + ab - ac}{-a}$$

Now, we need to divide each term of the polynomial by the monomial and then simplify.

= $$\frac{a^{2}}{-a} + \frac{ab}{-a} - \frac{ac}{-a}$$

= $$-\frac{a^{2}}{a} - \frac{ab}{a} + \frac{ac}{a}$$

Now each term will be simplified by cancelling out the common factor.

= -a - b + c

3. Find the quotient a3 - a2b – a2b2 by a2

= a3 - a2b – a2b2 ÷ a2

= $$\frac{a^{3} - a^{2}b - a^{2}b^{2}}{a^{2}}$$

Now, we need to divide each term of the polynomial by the monomial and then simplify.

= $$\frac{a^{3}}{a^{2}} - \frac{a^{2}b}{a^{2}} - \frac{a^{2}b^{2}}{a^{2}}$$

Now each term will be simplified by cancelling out the common factor.

= a - b - b2

4. Find the quotient 4m4n4 – 8m3n4 + 6mn3 by -2mn

= 4m4n4 – 8m3n4 + 6mn3 ÷ -2mn

= $$\frac{4m^{4}n^{4} - 8m^{3}n^{4} + 6mn^{3}}{-2mn}$$

Now, we need to divide each term of the polynomial by the monomial and then simplify.

= $$\frac{4m^{4}n^{4}}{-2mn} - \frac{8m^{3}n^{4}}{-2mn} + \frac{6mn^{3}}{-2mn}$$

= $$-\frac{4m^{4}n^{4}}{2mn} + \frac{8m^{3}n^{4}}{2mn} - \frac{6mn^{3}}{2mn}$$

Now each term will be simplified by cancelling out the common factor.

= 2m3n3 + 4m2n3 - 3n2

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