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QuestionSimplify: 10m3n4−15mn5mn\frac{10 m^{3} n^{4}-15 m n}{5 m n} and identify the results of each expression.

Studdy Solution

STEP 1

Assumptions1. We are given the expression 10m3n4−15mn5mn\frac{10 m^{3} n^{4}-15 m n}{5 m n}. . We are asked to divide this expression by the following polynomials m4n5−3mn m^{4} n^{5}-3 m^{} n^{}, 5mn3−105 m^{} n^{3}-10, mn3−15mn m^{} n^{3}-15 m n, and mn3−3 m^{} n^{3}-3.

STEP 2

First, let's simplify the given expression. We can do this by factoring out the common factor of 5mn5mn from the numerator.
10mn4−15mn5mn=5mn(2m2n−)5mn\frac{10 m^{} n^{4}-15 m n}{5 m n} = \frac{5mn(2m^{2}n^{}-)}{5 m n}

STEP 3

Now, cancel out the common factor of 5mn5mn from the numerator and the denominator.
5mn(2m2n3−3)5mn=2m2n3−3\frac{5mn(2m^{2}n^{3}-3)}{5 m n} =2m^{2}n^{3}-3

STEP 4

Now, let's divide the simplified expression by each of the given polynomials.First, divide by 2m4n−3m2n22 m^{4} n^{}-3 m^{2} n^{2}2m2n3−32m4n−3m2n2\frac{2m^{2}n^{3}-3}{2 m^{4} n^{}-3 m^{2} n^{2}}

STEP 5

Next, divide by 5m2n3−105 m^{2} n^{3}-102m2n3−35m2n3−10\frac{2m^{2}n^{3}-3}{5 m^{2} n^{3}-10}

STEP 6

Then, divide by 2m2n3−15mn2 m^{2} n^{3}-15 m n2m2n3−32m2n3−15mn\frac{2m^{2}n^{3}-3}{2 m^{2} n^{3}-15 m n}

STEP 7

Finally, divide by 2m2n3−32 m^{2} n^{3}-32m2n3−32m2n3−3\frac{2m^{2}n^{3}-3}{2 m^{2} n^{3}-3}

STEP 8

The last expression simplifies to1, because any number or expression divided by itself is equal to1.2m2n3−32m2n3−3=1\frac{2m^{2}n^{3}-3}{2 m^{2} n^{3}-3} =1So, the results of the divisions are1. 2m2n3−32m4n5−3m2n2\frac{2m^{2}n^{3}-3}{2 m^{4} n^{5}-3 m^{2} n^{2}}
2. 2m2n3−35m2n3−10\frac{2m^{2}n^{3}-3}{5 m^{2} n^{3}-10}
3. 2m2n3−32m2n3−15mn\frac{2m^{2}n^{3}-3}{2 m^{2} n^{3}-15 m n}
4. 11

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