Question

If \[x + a\] is the factor of \[{x^4} - {a^2}{x^2} + 3x - 6a\] then, \[a = \]
A. 0
B. -1
C. 1
D. 2

Answer 1

Hint: Remainder theorem is used to determine the remainder when two polynomials are given in the question for the division. The remainder theorem states that when a polynomial\[g\left( x \right)\] is divided by a linear polynomial, \[\left( {x - a} \right)\] then we get the remainder as \[g(a)\]and quotient as $q(x)$. The remainder is an integer that is left behind when a dividend is divided by the divisor. This theorem is applicable only when the polynomial is divided by a linear polynomial. A linear polynomial is generally represented as \[f\left( x \right) = ax + b\]where \[b\] is the constant term, whereas polynomial is the expression which consists of variables and coefficients having the operations of addition, multiplication, subtraction, and exponent. A polynomial function in degree n can be written as \[g(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ........... + {a_2}{x^2} + {a_1}{x^1} + {a_0}\].
Another method to find the remainder is by dividing the dividend by the divisor, which is the general method to find the remainder, \[{\text{R = }}\dfrac{{{\text{dividend}}}}{{{\text{divisor}}}}\]

Complete step by step solution: Here \[{x^4} - {a^2}{x^2} + 3x - 6a\] is the dividend to which number is to be divided and \[x + a\] is the divisor by which the number is divided.
Now equate the divisor, which is a linear polynomial with \[0\] hence we get,
\[
  x + a = 0 \\
  x = - a \\
 \]
So we can conclude when the polynomial \[f\left( x \right) = {x^4} - {a^2}{x^2} + 3x - 6a\] is divided by the linear polynomial, \[x = - a\]we get the remainder\[f\left( { - a} \right)\],
\[
  f\left( x \right) = {x^4} - {a^2}{x^2} + 3x - 6a \\
  f\left( { - a} \right) = {\left( { - a} \right)^4} - {a^2}{\left( { - a} \right)^2} + 3\left( { - a} \right) - 6a \\
   = {a^4} - {a^4} - 3a - 6a \\
   = - 9a \\
 \]
We know it \[\left( {x + a} \right)\] is the factor of \[f\left( x \right)\]then we can write \[f\left( { - a} \right) = 0\] ; hence we get
\[
  f\left( { - a} \right) = 0 \\
   - 9a = 0 \\
  a = 0 \\
 \]
So the value of a is 0.

Option A is correct.

Note: Before finding the remainder, check whether the divisor is a linear polynomial; otherwise, the division is not possible. Another method to find the remainder is by dividing the dividend by the divisor by a long division method, which includes more calculations, and the chances of getting the error are more.