Question

What is the value of b when we are given that $P\left( x \right)$ is divided by $D\left( x \right)$ and $R\left( x \right)$ is the remainder?
$P\left( x \right) = {x^4} - {x^3} - 3b{x^2} + 12{b^4}$
$D\left( x \right) = \left( {x + 3b} \right)$
$R\left( x \right) = 93$

Answer 1

Hint: We are required to find the value of the variable b in the function $P\left( x \right)$ with the help of a given division process. This question requires us to have the knowledge of basic and simple algebraic rules and operations such as substitution, addition, multiplication, subtraction and many more like these. A thorough understanding of functions, division algorithms and its applications will be of great significance.

Complete step-by-step answer:
In the given question, we are required to find the value of variable b in the function $P\left( x \right)$.
So, we are given the polynomial $P\left( x \right)$ as dividend, $D\left( x \right)$as divisor and $R\left( x \right)$ as remainder.
Now, we know that we can evaluate the remainder using the remainder theorem by substituting the value of variable in dividend.
First equating the divisor $D\left( x \right)$ as zero, we get the value of x as,
$ \Rightarrow D\left( x \right) = \left( {x + 3b} \right) = 0$
$ \Rightarrow x = - 3b$
So, we substitute the value of x as $\left( { - 3b} \right)$ in the original dividend function to get the remainder using the remainder theorem.
Hence, we get the remainder as
$ \Rightarrow P\left( { - 3b} \right) = {\left( { - 3b} \right)^4} - {\left( { - 3b} \right)^3} - 3b{\left( { - 3b} \right)^2} + 12{b^4}$
Now, evaluating the powers of $\left( { - 3b} \right)$, we get,
$ \Rightarrow P\left( { - 3b} \right) = 81{b^4} - \left( { - 27{b^3}} \right) - 3b\left( {9{b^2}} \right) + 12{b^4}$
Opening the brackets, we get,
\[ \Rightarrow P\left( { - 3b} \right) = 81{b^4} + 27{b^3} - 27{b^3} + 12{b^4}\]
Cancelling the like terms with opposite signs, we get,
\[ \Rightarrow P\left( { - 3b} \right) = 81{b^4} + 12{b^4}\]
Simplifying further, we get,
\[ \Rightarrow P\left( { - 3b} \right) = 93{b^4}\]
Now, we also know that the remainder obtained on dividing the polynomial $P\left( x \right)$ by the divisor $D\left( x \right)$ is $R\left( x \right) = 93$.
So, equating both, we get,
$ \Rightarrow 93{b^4} = 93$
Dividing both sides of the equation by $93$ and finding the value of b, we get,
$ \Rightarrow {b^4} = \dfrac{{93}}{{93}} = 1$
$ \Rightarrow b = 1$
So, the value of b is $1$.
So, the correct answer is “1”.

Note: Remainder theorem is an approach of Euclidean division of polynomials. Remainder theorem requires just a simple change of variable in the function so as to find the remainder of the division procedure. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable.