Dividing Polynomials by Monomial

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To multiply polynomial by monomial, one should use sum rule for fractions in reverse direction: `(a+b)/c=a/c+b/c` (in fact, there can be more than two terms in the numerator).

In other words, we just break down polynomial, then, just divide monomials and you're done.

Example 1. Divide `(4x^3+6x^2+10x)/(2x)`.

`color(green)(4x^3+6x^2+10x)/(color(red)(2x))=`

`=(color(green)(4x^3))/(color(red)(2x))+(color(green)(6x^2))/(color(red)(2x))+(color(green)(10x))/(color(red)(2x))=` (break down polynomial)

`=2x^2+3x+5` (divide monomials)

Answer: `(4x^3+6x^2+10x)/(2x)=2x^2+3x+5`.

Negative terms are handled in the same way.

Example 2. Multiply the following: `(3x^3-x^2-5x+3/7)-:(1/3x^2)`.

`(color(green)(2x^3-x^2-5x+3/7))/(color(red)(1/3x^2))=`

`=(color(green)(2x^3))/(color(red)(1/3x^2))+(color(green)(-x^2))/(color(red)(1/3x^2))+(color(green)(-5x))/(color(red)(1/3x^2))+(color(green)(3/7))/(color(red)(1/3x^2))=` (split polynomial)

`=6x+3-15/x+9/(7x^2)` (divide monomials)

Answer: `(3x^3-x^2-5x+3/7)-:(1/3x^2)=6x+3-15/x+9/(7x^2)`.

Note: above example shows, that result of division polynomial by monomial is not always polynomial.

Of course, polynomials with many variables can also be handled in a similar way.

Example 3. Divide `(3x^5y+2xz-7xy^2-z)` by `-4x^2y^2`.

`(3x^5y+2xz-7xy^2-z)/(-4x^2y^2)=`

`=(3x^5y)/(-4x^2y^2)+(2xz)/(-4x^2y^2)+(-7xy^2)/(-4x^2y^2)+(-z)/(-4x^2y^2)=`

`=-(3x^3)/(4y)-z/(2xy^2)+7/(4x)+z/(4x^2y^2)`.

Answer: `(3x^5y+2xz-7xy^2-z)/(-4x^2y^2)=-(3x^3)/(4y)-z/(2xy^2)+7/(4x)+z/(4x^2y^2)`.

Now, it is time to exercise.

Exercise 1. Divide `(x^5+2x^4+5x^2)/(5x^2)`.

Answer: `1/5x^3+2/5x^2+1`.

Exercise 2. Divide `(3a^3-7a^2+2b)` by `-2/7a^2`.

Answer: `-21/2a+49/2-(7b)/(a^2)`.

Exercise 3. Divide the following: `(5a^2b^2-3a^3bc+3/5a^2b-1/10ab)-:(3ab)`.

Answer: `5/3ab-a^2c+1/5a-1/30`.