Based On The Polynomial Remainder Theorem, What Is The Value Of The Function When $x=5$? F ( X ) = X 4 − 2 X 3 + 5 X 2 − 7 X + 4 F(x) = X^4 - 2x^3 + 5x^2 - 7x + 4 F ( X ) = X 4 − 2 X 3 + 5 X 2 − 7 X + 4 Enter Your Answer In The Box.$f(5) = $ □ \square □

Introduction

The polynomial remainder theorem is a fundamental concept in algebra that helps us find the value of a polynomial function at a specific point. This theorem is based on the idea that if we divide a polynomial by a linear factor, the remainder will be the value of the polynomial at the root of that factor. In this article, we will explore the polynomial remainder theorem and use it to find the value of the function $f(x) = x^4 - 2x^3 + 5x^2 - 7x + 4$ when $x=5$.

What is the Polynomial Remainder Theorem?

The polynomial remainder theorem states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder will be the value of the polynomial at $x=a$. Mathematically, this can be expressed as:

$$f(a) = r$$

where $r$ is the remainder.

How to Apply the Polynomial Remainder Theorem

To apply the polynomial remainder theorem, we need to follow these steps:

1. Divide the polynomial by the linear factor: We need to divide the polynomial $f(x)$ by the linear factor $(x-a)$.
2. Find the remainder: The remainder will be the value of the polynomial at $x=a$.

Example: Finding the Value of a Function using the Polynomial Remainder Theorem

Let's use the polynomial remainder theorem to find the value of the function $f(x) = x^4 - 2x^3 + 5x^2 - 7x + 4$ when $x=5$.

Step 1: Divide the polynomial by the linear factor

We need to divide the polynomial $f(x) = x^4 - 2x^3 + 5x^2 - 7x + 4$ by the linear factor $(x-5)$.

import sympy as sp
x = sp.symbols('x')
f = x4 - 2*x3 + 5x**2 - 7x + 4

remainder = sp.rem(f, x-5)
print(remainder)

Step 2: Find the remainder

The remainder will be the value of the polynomial at $x=5$.

# Evaluate the remainder at x=5
remainder_value = remainder.subs(x, 5)
print(remainder_value)

Conclusion

In this article, we used the polynomial remainder theorem to find the value of the function $f(x) = x^4 - 2x^3 + 5x^2 - 7x + 4$ when $x=5$. We divided the polynomial by the linear factor $(x-5)$ and found the remainder, which is the value of the polynomial at $x=5$. The polynomial remainder theorem is a powerful tool for finding the value of a polynomial function at a specific point.

Final Answer

Introduction

In our previous article, we explored the polynomial remainder theorem and used it to find the value of the function $f(x) = x^4 - 2x^3 + 5x^2 - 7x + 4$ when $x=5$. In this article, we will answer some frequently asked questions about the polynomial remainder theorem.

Q&A

Q: What is the polynomial remainder theorem?

A: The polynomial remainder theorem is a fundamental concept in algebra that helps us find the value of a polynomial function at a specific point. It states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder will be the value of the polynomial at $x=a$.

Q: How do I apply the polynomial remainder theorem?

A: To apply the polynomial remainder theorem, you need to follow these steps:

1. Divide the polynomial by the linear factor: You need to divide the polynomial $f(x)$ by the linear factor $(x-a)$.
2. Find the remainder: The remainder will be the value of the polynomial at $x=a$.

Q: What is the difference between the polynomial remainder theorem and the factor theorem?

A: The polynomial remainder theorem and the factor theorem are related but distinct concepts. The factor theorem states that if $f(a) = 0$, then $(x-a)$ is a factor of $f(x)$. The polynomial remainder theorem, on the other hand, states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder will be the value of the polynomial at $x=a$.

Q: Can I use the polynomial remainder theorem to find the roots of a polynomial?

A: Yes, you can use the polynomial remainder theorem to find the roots of a polynomial. If you know that $(x-a)$ is a factor of $f(x)$, then you can use the polynomial remainder theorem to find the value of the polynomial at $x=a$. If the value is zero, then $a$ is a root of the polynomial.

Q: How do I use the polynomial remainder theorem with complex numbers?

A: The polynomial remainder theorem works with complex numbers just like it works with real numbers. If you need to find the value of a polynomial at a complex number, you can use the polynomial remainder theorem as usual.

Q: Can I use the polynomial remainder theorem to find the value of a polynomial at a non-integer value?

A: Yes, you can use the polynomial remainder theorem to find the value of a polynomial at a non-integer value. If you need to find the value of a polynomial at a non-integer value, you can use the polynomial remainder theorem as usual.

Q: Is the polynomial remainder theorem only for polynomials of degree 1?

A: No, the polynomial remainder theorem is not only for polynomials of degree 1. It works for polynomials of any degree.

Q: Can I use the polynomial remainder theorem to find the value of a polynomial at a very large or very small value?

A: Yes, you can use the polynomial remainder theorem to find the value of a polynomial at a very large or very small value. If you need to find the value of a polynomial at a very large or very small value, you can use the polynomial remainder theorem as usual.

Conclusion

In this article, we answered some frequently asked questions about the polynomial remainder theorem. We hope that this article has been helpful in clarifying any doubts you may have had about the polynomial remainder theorem.

Final Answer

The final answer is $\boxed{24}$.