Question

How do you simplify and divide \[\left( {{a^3}{b^2} - {a^2}b + 2a} \right){\left( { - ab} \right)^{ - 1}}\] ?

Answer 1

Hint: The polynomial division involves the division of one polynomial by another. The division of polynomials can be between two monomials, a polynomial and a monomial or between two polynomials. To simplify the given expression, we need to arrange the polynomial to be divided in the standard form i.e., with respect to dividend and divisor, hence by this we need to simplify and divide the given expression.

Complete step-by-step answer:
Given,
 \[\left( {{a^3}{b^2} - {a^2}b + 2a} \right){\left( { - ab} \right)^{ - 1}}\]
Let us arrange the polynomial to be divided in the standard form, hence the given equation is rewritten as:
 \[ \Rightarrow \dfrac{{{a^3}{b^2} - {a^2}b + 2a}}{{ - ab}}\]
Taking out ‘a’ as a common term, we have:
 \[ \Rightarrow \dfrac{{a\left( {{a^2}{b^2} - ab + 2} \right)}}{{ - ab}}\]
As, numerator and denominator consist of same term i.e., ‘a’, hence combining and simplifying the terms, we get:
 \[ \Rightarrow - \dfrac{{{a^2}{b^2} - ab + 2}}{b}\]
Now, let us further divide the terms, we get:
 \[ = - \dfrac{{{a^2}{b^2}}}{b} + \dfrac{{ab}}{b} - \dfrac{2}{b}\]
 \[ = - {a^2}b + a - \dfrac{2}{b}\]
Therefore, we have:
 \[\left( {{a^3}{b^2} - {a^2}b + 2a} \right){\left( { - ab} \right)^{ - 1}} = - {a^2}b + a - \dfrac{2}{b}\]
So, the correct answer is “ \[ - {a^2}b + a - \dfrac{2}{b}\] ”.

Note: For dividing a polynomial with another polynomial, the polynomial is written in standard form i.e., the terms of the dividend and the divisor are arranged in decreasing order of their degrees. It is to be noted that the highest power(degree) of the polynomial gives the maximum number of zeros of the polynomial.