Question

For ${x^2} + 2x + 5$ to be a factor of ${x^4} + p{x^2} + q$, the values of $p$ and $q$ must be, respectively.
A) $ - 2,5$
B) 5,25
C) 10,20
D) 6,25
E) 14,25

Answer 1

Hint: Divide the polynomial ${x^4} + p{x^2} + q$ with its factor ${x^2} + 2x + 5$ and find the remainder. If a polynomial is divided by its factor, then the remainder is equal to zero. Hence, equate the calculated remainder to zero and find the value of unknowns.

Complete step by step solution: Since, we are given that ${x^2} + 2x + 5$ be a factor of ${x^4} + p{x^2} + q$, then, if we will divide ${x^4} + p{x^2} + q$ by ${x^2} + 2x + 5$ , then the remainder will be zero.
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 \\
$

Thus option D is the correct answer.

Note: A polynomial is an algebraic expression is such as ${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ....{a_2}{x^2} + {a_1}x + {a_0}$,where \[n\] is the degree of polynomial when ${a_n} \ne 0$.When a polynomial is divided by its factor, then the remainder is zero. Also, the zeroes of the factor will be the zeroes of the polynomial. Divide the polynomial correctly to avoid incorrect answers.