Calculator Based on the polynomial remainder theorem, what is the value of the function when x=5 ? f(x)=x^(4)-2x^(3)+5x^(2)-7x+4 explain it briefly and calculate the function from given data
The Correct Answer and Explanation is :
The Polynomial Remainder Theorem states that for a polynomial ( f(x) ), when you divide ( f(x) ) by ( (x – c) ), the remainder of this division is equal to ( f(c) ). In simpler terms, to evaluate the value of a polynomial at any specific point ( x = c ), you substitute ( c ) into the polynomial equation. This theorem is particularly useful because it avoids the need to actually divide the polynomial, allowing you to quickly find the value of the function at that point.
The Problem
We are given the polynomial:
[
f(x) = x^4 – 2x^3 + 5x^2 – 7x + 4
]
We are asked to find the value of ( f(x) ) when ( x = 5 ). According to the Polynomial Remainder Theorem, to find ( f(5) ), all we need to do is substitute ( x = 5 ) into the polynomial expression.
Steps:
- Substitute ( x = 5 ) into the equation:
[
f(5) = 5^4 – 2(5^3) + 5(5^2) – 7(5) + 4
]
- Simplify each term:
- ( 5^4 = 625 )
- ( 5^3 = 125 ), so ( -2(5^3) = -2 \times 125 = -250 )
- ( 5^2 = 25 ), so ( 5(5^2) = 5 \times 25 = 125 )
- ( 7(5) = 35 )
- Now, substitute all the simplified values:
[
f(5) = 625 – 250 + 125 – 35 + 4
]
- Perform the arithmetic:
[
f(5) = 625 – 250 = 375
]
[
375 + 125 = 500
]
[
500 – 35 = 465
]
[
465 + 4 = 469
]
Final Answer:
The value of ( f(x) ) when ( x = 5 ) is ( \boxed{469} ).
Explanation:
To solve this, we simply used the Polynomial Remainder Theorem, which allowed us to evaluate the function directly by plugging in ( x = 5 ) into the polynomial. We then performed basic arithmetic operations (addition, subtraction, and multiplication) to simplify the expression and arrive at the final value, ( 469 ). This method is quick and efficient, particularly when you’re dealing with polynomials of higher degrees.