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

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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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