Questions & Answers

Question

Answers

A) $ - 2,5$

B) 5,25

C) 10,20

D) 6,25

E) 14,25

Answer
Verified

A factor divides the number completely.

Hence, we will find the remainder after dividing ${x^4} + p{x^2} + q$ by ${x^2} + 2x + 5$ and equate that remainder to zero to find the value of unknowns.

After, dividing we get the remainder as \[\left( {12 - 2p} \right)x + \left( {q - 5p + 5} \right)\]

Now, we will equate the remainder to be 0.

The coefficient of the $x$ should be 0 and the constant term of the remainder must be 0 for the complete remainder to be zero.

Equating the coefficient of $x$ to 0 and solve for the value of $p$

$

12 - 2p = 0 \\

\Rightarrow 2p = 12 \\

\Rightarrow p = 6 \\

$

Equating the constant term of the remainder to 0 , we get

\[q - 5p + 5 = 0\]

Substituting the value 6 for $p$ in the above equation and solve for the value of $q$

$

q - 5\left( 6 \right) + 5 = 0 \\

\Rightarrow q - 30 + 5 = 0 \\

\Rightarrow q = 25 \\

$