Question & Answer
QUESTION

If one factor of ${a^4} + {\text{ }}{b^4} + {\text{ }}{a^2}{b^2}$ is ${a^2} + {\text{ }}{b^2} + {\text{ }}ab$, then the other factor is

A. ${a^3} + {b^3} + {c^3}$

B. ${a^2} + {b^2} - ab$

C. ${a^2} + {b^2} + {c^2}$ 

D. ${a^2} + {b^2} + ab$

ANSWER Verified Verified

We know that, if we are provided with a factor of a number and we have to find another factor we simply divide the number by the given factor.

For example we know that 3 is one of a factor of 12 and we have to find another factor we simply divide

12 by 3 to know another factor i.e, $\dfrac{{12}}{3} = 4$

Thus, 4 is another factor of 12. 

In the similar way we will find the factor of given question

Given that ${a^2} + {b^2} + ab{\text{ is a factor of }}{a^4} + {b^4} + {a^2}{b^2}$ 

Now another factor is determined by following method:

$

{a^2} + {b^2} + ab\mathop{\left){\vphantom{1\begin{gathered}

  {a^4} + {b^4} + {a^2}{b^2} 

   - {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

  {b^4}{\text{ }} - {a^3}b 

  {b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

   - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

   - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

 }}0 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_  

\end{gathered} }}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}

  {a^4} + {b^4} + {a^2}{b^2} 

   - {a^2}{\text{ - }}{a^2}{b^2}{\text{ - }}{a^3}b 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

  {b^4}{\text{ }} - {a^3}b 

  {b^4}{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

   - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

   - {a^3}b{\text{ }} - a{b^3}{\text{ }} - {a^2}{b^2} 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 

 }}0 

  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_  

\end{gathered} }}}

\limits^{\displaystyle \,\,\, {{a^2} + {b^2} - ab}}

$

Thus the another factor is {a^2} + {b^2} - ab 

Hence, the correct option is (b)

 

Note: - In these types of questions we simply divide the given factor with the given polynomial.