Question

One factor of \[{{x}^{4}}+\text{ }{{x}^{2}}\text{-20 }is\text{ }{{x}^{2}}+5\] . The other factor is:-
(a)\[{{x}^{2}}-4\]
(b)\[x-4\]
(c)\[{{x}^{2}}-5\]
(d)\[x+2\]

Answer 1

Hint: Before solving this question, let us know about factors and multiples.
Factors: Factors are the numbers we multiply together to get another number: There can be many factors of a number.
Multiples: A number that can be divided completely by another number is a multiple of that number.

Complete step-by-step answer:
Let us take an example. Say, 12.
Factors of 12:-
\[\begin{align}
  & 1~\times 12 \\
 & 2~\times 6 \\
 & 3\times ~4 \\
\end{align}\]
Let us take an example. Say 4 is a multiple of 2, because the number 4 is completely divided by 2.
According to the question, one of the factors of \[({{x}^{4}}+\text{ }{{x}^{2}}-20)\ is\text{ }\left( {{x}^{2}}+\text{ }5 \right)\] .
We need to find the other factor.
Let the other factor of \[({{x}^{4}}+\text{ }{{x}^{2}}-20)\] be ‘a’.
So, when we will multiply ‘a’ by \[({{x}^{2}}\text{+ 5)}\] , then the product must be \[({{x}^{4}}+\text{ }{{x}^{2}}-20)\]
Let us now solve this question. We shall consider every option.
\[{{x}^{2}}-4\]
Let us check that when we multiply \[{{x}^{2}}-4\] with \[{{x}^{2}}+5\] , then whether we get \[({{x}^{4}}+\text{ }{{x}^{2}}-20)\] or not.
\[\begin{array}{*{35}{l}}
   \left( {{x}^{2}}-4 \right)\cdot \left( {{x}^{2}}+\text{ }5 \right)\text{ }=\text{ }{{x}^{2}}\left( {{x}^{2}}+\text{ }5 \right)-4\left( {{x}^{2}}+\text{ }5 \right) \\
   ~~~~~~~~~~~~~~~~~~~~~~~~~~\ \ \ \ \ \ =\text{ }{{x}^{4}}+\text{ }5{{x}^{2}}-4{{x}^{2}}-20 \\
   ~~~~~~~~~~~~~~~~~~~~~~~~~~\ \ \ \ \ \ =\text{ }{{x}^{4}}+\text{ }{{x}^{2}}-20 \\
\end{array}\]
As we can see that the result of our explanation above matches \[({{x}^{4}}+\text{ }{{x}^{2}}-20)\]
\[({{x}^{4}}+{{x}^{2}}-20)=({{x}^{4}}+{{x}^{2}}-20)\]
Hence, verified
Therefore, the answer of this question is (a) \[{{x}^{2}}-\text{ }4\] .

Note: One must do all the calculations in this question very carefully.
Also not only in this question, the students must be very careful while solving any such questions as if there is any mistake in the calculus, then the answer can come out to be wrong.