Question

What is the quotient when $({{a}^{4}}-{{b}^{4}})$ is divided by $a-b?$
$\begin{align}
  & A.\text{ }{{\text{a}}^{\text{3}}}\text{+}{{\text{a}}^{\text{2}}}\text{b+a}{{\text{b}}^{\text{2}}}\text{+}{{\text{b}}^{\text{3}}} \\
 & B.\text{ }{{\text{a}}^{\text{3}}}\text{+}{{\text{a}}^{\text{2}}}\text{b+a}{{\text{b}}^{\text{2}}}\text{-}{{\text{b}}^{\text{3}}} \\
 & C.\text{ }{{\text{a}}^{\text{3}}}\text{+}{{\text{a}}^{\text{2}}}\text{b-a}{{\text{b}}^{\text{2}}}\text{+}{{\text{b}}^{\text{3}}} \\
 & D.\text{ }{{\text{a}}^{\text{3}}}\text{-}{{\text{a}}^{\text{2}}}\text{b+a}{{\text{b}}^{\text{2}}}\text{+}{{\text{b}}^{\text{3}}} \\
\end{align}$

Answer 1

Hint: In this question we have to find the quotient when we have to divide the polynomial $({{a}^{4}}-{{b}^{4}})$ by the polynomial $(a-b)$,
So first of all, we have to factorise $({{a}^{4}}-{{b}^{4}})$and then we divide these factors by $(a-b)$. During the process of division, we have to cancel out the common terms in order to find the quotient. We can factorise $({{a}^{4}}-{{b}^{4}})$ by using the formula ${{\alpha }^{2}}-{{\beta }^{2}}=(\alpha +\beta )(\alpha -\beta )$ and ${{\alpha }^{2n}}={{\left( {{\alpha }^{2}} \right)}^{n}}$

Complete step-by-step answer:
Let us first we factorize the given polynomial $({{a}^{4}}-{{b}^{4}})$.
So, we can write
$({{a}^{4}}-{{b}^{4}})={{\left( {{a}^{2}} \right)}^{2}}-{{\left( {{b}^{2}} \right)}^{2}}$
Here we use the formula ${{\alpha }^{2n}}={{\left( {{\alpha }^{2}} \right)}^{n}}$
Noe let us assume that ${{a}^{2}}=x,and\text{ }{{b}^{2}}=y$
So, we can write the above as
$({{a}^{4}}-{{b}^{4}})={{\left( x \right)}^{2}}-{{\left( y \right)}^{2}}$
As we know that ${{\alpha }^{2}}-{{\beta }^{2}}=(\alpha +\beta )(\alpha -\beta )$
So, we can write
$({{a}^{4}}-{{b}^{4}})=(x+y)(x-y)$
Now substituting the values of$x={{a}^{2}}\text{ and }y={{b}^{2}}$we can write
$({{a}^{4}}-{{b}^{4}})=({{a}^{2}}+{{b}^{2}})({{a}^{2}}-{{b}^{2}})$
Again, we can factorise the polynomial $\left( {{a}^{2}}-{{b}^{2}} \right)$as
$\left( {{a}^{2}}-{{b}^{2}} \right)=(a+b)(a-b)$
So finally, we can write the polynomial$({{a}^{4}}-{{b}^{4}})$in its factorised form as
$({{a}^{4}}-{{b}^{4}})=({{a}^{2}}+{{b}^{2}})(a-b)(a+b)$
As we have to find the quotient so we divide $({{a}^{4}}-{{b}^{4}})$by $(a-b)$so we can write
$\left( \dfrac{{{a}^{4}}-{{b}^{4}}}{a-b} \right)=\dfrac{\left( {{a}^{2}}+{{b}^{2}} \right)(a+b)(a-b)}{\left( a-b \right)}$
Now cancelling the common terms, we can write
$\left( \dfrac{{{a}^{4}}-{{b}^{4}}}{a-b} \right)=({{a}^{2}}+{{b}^{2}})(a-b)$
Hence the quotient is $({{a}^{2}}+{{b}^{2}})(a-b)$
Now we have to expand the above quotient by multiplying the factors so we can write
\[\begin{align}
  & ({{a}^{2}}+{{b}^{2}})(a+b) \\
 & ={{a}^{2}}(a+b)+{{b}^{2}}(a+b) \\
 & ={{a}^{3}}+{{a}^{2}}b+{{b}^{2}}a+{{b}^{3}} \\
\end{align}\]
Now we see that in options that option B is correct.

Note: it should be noted that that we can find the quotient by long division method by following the rules below: -
A. Arrange divisor and dividend in ascending or descending powers of some common letter.
B. Divide the term on the left of the dividend by the term on the left of the divisor, and put the result in the quotient.
C. Multiply the WHOLE divisor by this quotient, and put the product under the dividend.
D. Subtract and bring down form the dividend as many terms as may be necessary.
Report these operations till all the terms from the dividend are brought down.