Question

# Perform the indicated divisions of polynomials by monomials. frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}}

Polynomial division
Perform the indicated divisions of polynomials by monomials.
$$\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}}$$

2021-02-20
A polynomial is an expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power.
Here the given polynomial is a trinomial.
To divide a polynomial by monomial, divide each term of the polynomial by the monomial.
Divide the trinomial by the monomial $$9a^{2}b^{2}$$.
Simplify the terms which are under division.
Calculation:
Consider the polynomial $$\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}}$$.
Divide each term of the polynomial by the monomial $$9a^{2}b^{2}$$.
$$\frac{-27a^{3}b^{4}-36a^{2}b^{3}+72a^{2}b^{5}}{9a^{2}b^{2}} = (-27a^{3}\frac{b^{4}}{9}a^{2}b^{2})+(-36a^{2}\frac{b^{3}}{9}a^{2}b^{2})+(72a^{2}\frac{b^{5}}{9}a^{2}b^{2})=$$
$$= 3ab^{2}-4b+8b^{3}$$
The simplified value of the polynomial is $$3ab^{2}-4b+8b^{3}$$.
Final statement:
The simplified value of the polynomial after division is equals to $$3ab^{2}-4b+8b^{3}$$.