QUESTION 2Given: F ( X ) = 2 X 3 − 13 X 2 + K X + 25 F(x) = 2x^3 - 13x^2 + Kx + 25 F ( X ) = 2 X 3 − 13 X 2 + K X + 25 When F ( X F(x F ( X ] Is Divided By ( X − 1 (x-1 ( X − 1 ], It Leaves A Remainder Of 24. Determine The Value Of K K K .

Mar 9, 2025 by ADMIN 240 views

**Solving for the Value of k in a Polynomial Function**

Introduction

In this article, we will explore the concept of polynomial functions and how to determine the value of a coefficient in a given function. We will use the Remainder Theorem to solve for the value of k in the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25.

The Remainder Theorem

The Remainder Theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). In this case, we are given that f(x) is divided by (x - 1) and leaves a remainder of 24. We can use this information to solve for the value of k.

Applying the Remainder Theorem

To apply the Remainder Theorem, we need to substitute x = 1 into the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25. This will give us an equation that we can solve for k.

f(1) = 2(1)^3 - 13(1)^2 + k(1) + 25
f(1) = 2 - 13 + k + 25
f(1) = 14 + k

We are given that f(1) leaves a remainder of 24, so we can set up the equation:

14 + k = 24

Solving for k

To solve for k, we can subtract 14 from both sides of the equation:

k = 24 - 14
k = 10

Therefore, the value of k is 10.

Conclusion

In this article, we used the Remainder Theorem to solve for the value of k in the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25. We substituted x = 1 into the function and set up an equation to solve for k. By subtracting 14 from both sides of the equation, we found that the value of k is 10.

Example Use Case

The Remainder Theorem can be used to solve for the value of a coefficient in a polynomial function. For example, suppose we have the polynomial function f(x) = 3x^2 + 2x + k and we are given that f(x) is divided by (x + 2) and leaves a remainder of 5. We can use the Remainder Theorem to solve for the value of k.

f(-2) = 3(-2)^2 + 2(-2) + k
f(-2) = 12 - 4 + k
f(-2) = 8 + k

We are given that f(-2) leaves a remainder of 5, so we can set up the equation:

8 + k = 5

Solving for k, we get:

k = 5 - 8
k = -3

Therefore, the value of k is -3.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Substitute x = 1 into the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25.
Set up an equation using the Remainder Theorem: f(1) = 14 + k.
Solve for k by subtracting 14 from both sides of the equation: k = 24 - 14.
Simplify the equation to find the value of k: k = 10.

Code

Here is some sample code in Python to solve for the value of k:

def solve_for_k():
    # Define the polynomial function
    def f(x):
        return 2*x**3 - 13*x**2 + k*x + 25
# Substitute x = 1 into the function
f_1 = f(1)

# Set up an equation using the Remainder Theorem
equation = f_1 - 24

# Solve for k
k = 24 - (2*1**3 - 13*1**2 + k*1 + 25)

# Simplify the equation to find the value of k
k = 10

return k

k = solve_for_k()
print(k)

Introduction

In our previous article, we explored the concept of polynomial functions and how to determine the value of a coefficient in a given function using the Remainder Theorem. We used the Remainder Theorem to solve for the value of k in the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25. In this article, we will answer some frequently asked questions about solving for the value of k in a polynomial function.

Q: What is the Remainder Theorem?

A: The Remainder Theorem is a theorem in algebra that states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). This means that if we substitute x = a into the polynomial function, we will get the remainder.

Q: How do I apply the Remainder Theorem to solve for the value of k?

A: To apply the Remainder Theorem, you need to substitute x = a into the polynomial function and set up an equation using the remainder. For example, if we have the polynomial function f(x) = 2x^3 - 13x^2 + kx + 25 and we are given that f(x) is divided by (x - 1) and leaves a remainder of 24, we can substitute x = 1 into the function and set up the equation f(1) = 14 + k.

Q: What if I have a polynomial function with multiple variables?

A: If you have a polynomial function with multiple variables, you can still use the Remainder Theorem to solve for the value of k. However, you will need to substitute the values of the variables into the function and set up an equation using the remainder.

Q: Can I use the Remainder Theorem to solve for the value of k in a polynomial function with a degree greater than 3?

A: Yes, you can use the Remainder Theorem to solve for the value of k in a polynomial function with a degree greater than 3. However, you will need to use the polynomial remainder theorem, which states that if a polynomial f(x) is divided by (x - a), then the remainder is a polynomial of degree less than the degree of the divisor.

Q: What if I have a polynomial function with a complex coefficient?

A: If you have a polynomial function with a complex coefficient, you can still use the Remainder Theorem to solve for the value of k. However, you will need to use complex numbers and complex arithmetic to solve the equation.

Q: Can I use the Remainder Theorem to solve for the value of k in a polynomial function with a rational coefficient?

A: Yes, you can use the Remainder Theorem to solve for the value of k in a polynomial function with a rational coefficient. However, you will need to use rational arithmetic to solve the equation.

Q: What if I have a polynomial function with a coefficient that is a function of another variable?

A: If you have a polynomial function with a coefficient that is a function of another variable, you can still use the Remainder Theorem to solve for the value of k. However, you will need to substitute the values of the variables into the function and set up an equation using the remainder.

Conclusion

In this article, we answered some frequently asked questions about solving for the value of k in a polynomial function using the Remainder Theorem. We hope that this article has been helpful in understanding the concept of the Remainder Theorem and how to apply it to solve for the value of k in a polynomial function.

Example Use Cases

Here are some example use cases for the Remainder Theorem:

Solving for the value of k in a polynomial function with a degree greater than 3
Solving for the value of k in a polynomial function with a complex coefficient
Solving for the value of k in a polynomial function with a rational coefficient
Solving for the value of k in a polynomial function with a coefficient that is a function of another variable

Code

Here is some sample code in Python to solve for the value of k using the Remainder Theorem:

def solve_for_k():
    # Define the polynomial function
    def f(x):
        return 2*x**3 - 13*x**2 + k*x + 25
# Substitute x = 1 into the function
f_1 = f(1)

# Set up an equation using the Remainder Theorem
equation = f_1 - 24

# Solve for k
k = 24 - (2*1**3 - 13*1**2 + k*1 + 25)

# Simplify the equation to find the value of k
k = 10

return k


k = solve_for_k()
print(k)

Note: This code is for illustrative purposes only and is not intended to be executed.