Question

How do you use the remainder theorem to find the remainder for the division $ 4{x^3} + 4{x^2} + 2x + 3 $ by $ \left( {x - 1} \right) $ ?

Answer 1

Hint: We are required to find the remainder of a function when it is divided by another linear function $ \left( {x - 1} \right) $ . 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 and remainder theorem, in specific, can be of great significance.

Complete step-by-step answer:
In the given question, we are required to find the remainder when $ 4{x^3} + 4{x^2} + 2x + 3 $ is divided by linear function $ \left( {x - 1} \right) $ .
According to the remainder theorem, we know that if a function is divided by linear polynomial function $ \left( {x - h} \right) $ , then we find the remainder by putting in the value of x as h in the function.
So, we need to replace the variable in the function given to us in the question by $ 1 $ so as to find the remainder when $ 4{x^3} + 4{x^2} + 2x + 3 $ is divided by $ \left( {x - 1} \right) $.
So, the function given to us is: $ f\left( x \right) = 4{x^3} + 4{x^2} + 2x + 3 $ .
We are required to find the value of $ f\left( 1 \right) $ by putting the value of variable x in the function.
Hence, $ f\left( x \right) = 4{x^3} + 4{x^2} + 2x + 3 $
 $ \Rightarrow f\left( 1 \right) = 4{\left( 1 \right)^3} + 4{\left( 1 \right)^2} + 2\left( 1 \right) + 3 $
 $ \Rightarrow f\left( 1 \right) = 4 + 4 + 2 + 3 $
 $ \Rightarrow f\left( 1 \right) = 13 $
Hence, we get the remainder for the division of $ 4{x^3} + 4{x^2} + 2x + 3 $ by $ \left( {x - 1} \right) $ as $ 13 $ .
So, the correct answer is “13”.

Note: Remainder theorem is a simple and easy way to find the remainder when a polynomial function is divided by another polynomial function. Such questions require simple change of variable and can be solved easily by keeping in mind the algebraic rules such as substitution and transposition. Substitution of a variable involves putting a certain value in place of the variable