# Dividing Polynomials by Monomial

## Related Calculator: Polynomial Calculator

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.