 If $x + a$ is the factor of ${x^4} - {a^2}{x^2} + 3x - 6a$ then, $a =$A. 0B. -1C. 1D. 2 Verified
146.4k+ views
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.