Question

How do you use the remainder theorem to evaluate \[f\left( x \right) = {x^5} - 47{x^3} - 16{x^2} + 8x + 52\] at \[x = 7\]?

Answer 1

Hint: Here, we will use the remainder theorem for the given polynomial by substituting the given value of the variable to find the required value. The Remainder theorem states that if a polynomial \[p\left( x \right)\] of degree greater than or equal to one is divided by a linear polynomial \[\left( {x - a} \right)\] then the remainder is \[p\left( a \right)\] where \[a\] is any real number.

Complete Step by Step Solution:
We are given a polynomial \[f\left( x \right) = {x^5} - 47{x^3} - 16{x^2} + 8x + 52\].
Now, we will evaluate the given polynomial at \[x = 7\].
By substituting \[x = 7\] in the given polynomial, we get
\[ \Rightarrow f\left( 7 \right) = {\left( 7 \right)^5} - 47{\left( 7 \right)^3} - 16{\left( 7 \right)^2} + 8\left( 7 \right) + 52\]
Applying the exponent on the terms, we get
\[ \Rightarrow f\left( 7 \right) = \left( {16807} \right) - 47\left( {343} \right) - 16\left( {49} \right) + 8\left( 7 \right) + 52\]
Multiplying the terms, we get
\[ \Rightarrow f\left( 7 \right) = \left( {16807} \right) - \left( {16121} \right) - \left( {784} \right) + \left( {56} \right) + 52\]
Now adding and subtracting the terms, we get
\[ \Rightarrow f\left( 7 \right) = 10\]

Therefore, the remainder of the given polynomial \[f\left( x \right) = {x^5} - 47{x^3} - 16{x^2} + 8x + 52\] at \[x = 7\] is 10.

Additional information:
We know that the BODMAS rule states that the first operation has to be done which is in the brackets, next the operation applies on the indices or order, then it moves on to the division and multiplication, and then using addition and subtraction we will simplify the expression. If addition or subtraction and division or multiplication is in the same calculations, then it has to be done from left to right.

Note:
We know that the Remainder theorem is useful in finding the remainder of the given polynomial without actually dividing the polynomial by the linear polynomial. The remainder can be zero or any positive integer but can never be a negative integer. The factor theorem is used for factoring and finding the solutions of polynomial equations and is the reverse of the remainder theorem.