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.
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 x^{6} + 7x^{5} – 5x^{4} by x^{2}= \(\frac{x^{6} + 7x^{5}  5x^{4}}{x^{2}}\)
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.
= \(x^{4} + 7x^{3}  5x^{2}\)
= \(\frac{a^{2} + ab  ac}{a}\)
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
= \(\frac{a^{3}  a^{2}b  a^{2}b^{2}}{a^{2}} \)
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  b^{2}= \(\frac{4m^{4}n^{4}  8m^{3}n^{4} + 6mn^{3}}{2mn}\)
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.
= 2m^{3}n^{3} + 4m^{2}n^{3}  3n^{2}7th Grade Math Problems
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