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One factor of ${x^4} + {x^2} - 20$ is ${x^2} + 5$. What is the other factor?
A. ${x^2} - 4$
B. $x - 4$
C. ${x^2} - 5$
D. $x + 2$

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Last updated date: 16th Jun 2024
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Answer
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Hint: Here we are given a factor of the polynomial. To find the other factor, we divide ${x^4} + {x^2} - 20$ by ${x^2} + 5$ using division algorithms and consider the quotient as the other factor.
Let’s take the number $10$ and one of its factors is $2$. To find the other factor, we divide $10$ by $2$, which gives the quotient as $5$. So, $5$ is another factor $10$.
We use the same logic for the given polynomials and find the required answer.

Complete step by step answer:
Step 1: We are given that ${x^2} + 5$ is a factor of${x^4} + {x^2} - 20$.
Now let, $p(x) = {x^4} + {x^2} - 20$ and $q(x) = {x^2} + 5$.
Since, $q(x)$ is a factor of $p(x)$. We know that $p(x) = q(x) \times g(x) + 0$, where $g(x)$ is the other factor. Here, $p(x)$ is the dividend, $q(x)$ is the divisor which is the given factor here, $g(x)$ is the quotient and remainder is zero. 

Step 2: Now to find $g(x)$, we should divide $p(x)$ by $q(x)$

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Here, p(x) is the dividend and q(x) is the divisor.


Step 3: Now, to start the division. First, we should divide the term with the highest degree of the dividend by the term with the highest degree of the divisor.
Here the term with the highest degree of the dividend is ${x^4}$and the term with the highest degree in the divisor is${x^2}$.
$ \Rightarrow \dfrac{{{x^4}}}{{{x^2}}} = {x^2}$
So, when we divide ${x^4}$ by ${x^2}$ we get${x^2}$, which is the first part of our quotient.

(i.e.) 
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Step 4: Multiply the term of the quotient which we found in the above step (i.e.)${x^2}$ with each term of the divisor and write it below the dividend
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Now, by subtracting ${x^4} + 5{x^2}$ from ${x^4} + {x^2} - 20$, we get
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Step 5: Now, let’s repeat steps 2 and 3 with our new dividend (i.e.) $ - 4{x^2} - 20$
In our new dividend, the term with the highest degree is and the term with the highest degree in the divisor is${x^2}$.
When we divide $ - 4{x^2}$ by ${x^2}$, we get$ - 4$.
$ \Rightarrow \dfrac{{ - 4{x^2}}}{{{x^2}}} = - 4$
This is the next term of our quotient.

Step 6: Now, multiply -4 with each term of the divisor and write it below the new dividend.
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When we subtract $ - 4{x^2} - 20$ from our new dividend we get 0.

Step 7: Now, the quotient obtained is the other factor.
Therefore, ${x^2} - 4$ is the other factor.


Note: While arranging the terms in the descending order of their degrees, write zero if any of the terms are missing as many students make a mistake while subtracting.
For example if the given polynomial is ${x^4} + {x^2} - 20$. Here the ${x^3}$ is missing so let’s write $0{x^3}$ as this will make the calculation easier.