Question

# 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. $x + 2$

Hint: The other factor is nothing other than the quotient when you divide ${x^4} + {x^2} - 20$ by ${x^2} + 5$ using division algorithm.

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, $p(x) = q(x)*g(x) + 0$, where g(x) is the other factor.
Step 2: Now to find g(x), we should divide p(x) by q(x)
${x^2} + 5\left){\vphantom{1{{x^4} + {x^2} + 20}}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{{{x^4} + {x^2} + 20}}}$ ……. (1)
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.) ${x^2} + 5\mathop{\left){\vphantom{1{{x^4} + {x^2} + 20}}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{{{x^4} + {x^2} + 20}}}} \limits^{\displaystyle \,\,\, {{x^2}}}$
Step 3: 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
${x^2} + 5\mathop{\left){\vphantom{1\begin{gathered} {x^4} + {x^2} - 20 \\ {x^4} + 5{x^2} \\ \end{gathered} }}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered} {x^4} + {x^2} - 20 \\ {x^4} + 5{x^2} \\ \end{gathered} }}} \limits^{\displaystyle \,\,\, {{x^2}}}$
Now, by subtracting ${x^4} + 5{x^2}$ from${x^4} + {x^2} - 20$, we get
${x^2} + 5\mathop{\left){\vphantom{1\begin{gathered} {x^4} + {x^2} - 20 \\ \underline {{x^4} + 5{x^2}} \\ - 4{x^2} - 20 \\ \end{gathered} }}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered} {x^4} + {x^2} - 20 \\ \underline {{x^4} + 5{x^2}} \\ - 4{x^2} - 20 \\ \end{gathered} }}} \limits^{\displaystyle \,\,\, {{x^2}}}$
Step 4: 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 5: Now, multiply -4 with each term of the divisor and write it below the new dividend.
${x^2} + 5\mathop{\left){\vphantom{1\begin{gathered} {x^4} + {x^2} - 20 \\ \underline {{x^4} + 5{x^2}} \\ - 4{x^2} - 20 \\ \underline { - 4{x^2} - 20} \\ 0 \\ \end{gathered} }}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered} {x^4} + {x^2} - 20 \\ \underline {{x^4} + 5{x^2}} \\ - 4{x^2} - 20 \\ \underline { - 4{x^2} - 20} \\ 0 \\ \end{gathered} }}} \limits^{\displaystyle \,\,\, {{x^2} - 4}}$
When we subtract $- 4{x^2} - 20$ from our new dividend we get 0.
Step 6: 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