Dividing the Polynomial Function

Oct 13, 2025 by talynt.org 31 views

Introduction

In algebra, it is often necessary to express a polynomial function in a specific form, such as (x-k)q(x) + r, where k is a given value. This form is particularly useful in various mathematical applications, including polynomial division and factoring. In this article, we will explore how to express the given polynomial function f(x) = 3x^4 + 4x^3 - 11x^2 + 16 in the form (x-k)q(x) + r, where k = -1.

Understanding the Form (x-k)q(x) + r

The form (x-k)q(x) + r represents a polynomial function that has been divided by (x-k). Here, q(x) is the quotient, and r is the remainder. The value of k is the root of the divisor (x-k). When a polynomial function f(x) is divided by (x-k), the result is a quotient q(x) and a remainder r, such that f(x) = (x-k)q(x) + r.

Dividing the Polynomial Function

To express the polynomial function f(x) = 3x^4 + 4x^3 - 11x^2 + 16 in the form (x-k)q(x) + r, we need to divide it by (x+1). We can use polynomial long division or synthetic division to perform this division.

Step 1: Set up the Division

To divide f(x) = 3x^4 + 4x^3 - 11x^2 + 16 by (x+1), we set up the division as follows:

  ____________________
x + 1 | 3x^4 + 4x^3 - 11x^2 + 16

Step 2: Divide the Leading Term

We divide the leading term of f(x), which is 3x^4, by the leading term of the divisor (x+1), which is x. This gives us 3x^3.

Step 3: Multiply and Subtract

We multiply the divisor (x+1) by the quotient term 3x^3, which gives us 3x^4 + 3x^3. We subtract this from the dividend f(x) = 3x^4 + 4x^3 - 11x^2 + 16.

  ____________________
x + 1 | 3x^4 + 4x^3 - 11x^2 + 16
  - (3x^4 + 3x^3)
  ____________________
  x^3 - 11x^2 + 16

Step 4: Repeat the Process

We repeat the process by dividing the leading term of the new dividend, which is x^3, by the leading term of the divisor (x+1), which is x. This gives us x^2.

Step 5: Multiply and Subtract

We multiply the divisor (x+1) by the quotient term x^2, which gives us x^3 + x^2. We subtract this from the new dividend x^3 - 11x^2 + 16.

  ____________________
x + 1 | 3x^4 + 4x^3 - 11x^2 + 16
  - (3x^4 + 3x^3)
  ____________________
  x^3 - 11x^2 + 16
  - (x^3 + x^2)
  ____________________
  -12x^2 + 16

Step 6: Repeat the Process

We repeat the process by dividing the leading term of the new dividend, which is -12x^2, by the leading term of the divisor (x+1), which is x. This gives us -12x.

Step 7: Multiply and Subtract

We multiply the divisor (x+1) by the quotient term -12x, which gives us -12x^2 - 12x. We subtract this from the new dividend -12x^2 + 16.

  ____________________
x + 1 | 3x^4 + 4x^3 - 11x^2 + 16
  - (3x^4 + 3x^3)
  ____________________
  x^3 - 11x^2 + 16
  - (x^3 + x^2)
  ____________________
  -12x^2 + 16
  - (-12x^2 - 12x)
  ____________________
  16 + 12x

Step 8: Write the Result

We have now completed the division of f(x) = 3x^4 + 4x^3 - 11x^2 + 16 by (x+1). The quotient is 3x^3 + x^2 - 12x, and the remainder is 16 + 12x.

Conclusion

In this article, we have expressed the polynomial function f(x) = 3x^4 + 4x^3 - 11x^2 + 16 in the form (x-k)q(x) + r, where k = -1. We used polynomial long division to divide f(x) by (x+1) and obtained the quotient 3x^3 + x^2 - 12x and the remainder 16 + 12x. This form is useful in various mathematical applications, including polynomial division and factoring.

Final Answer

The final answer is:

f(x) = (x+1)(3x^3 + x^2 - 12x) + (16 + 12x)

Introduction

In our previous article, we explored how to express a polynomial function in the form (x-k)q(x) + r, where k is a given value. We used polynomial long division to divide the polynomial function f(x) = 3x^4 + 4x^3 - 11x^2 + 16 by (x+1) and obtained the quotient 3x^3 + x^2 - 12x and the remainder 16 + 12x. In this article, we will answer some frequently asked questions related to expressing a polynomial function in the form (x-k)q(x) + r.

Q: What is the significance of the form (x-k)q(x) + r?

A: The form (x-k)q(x) + r is significant because it represents a polynomial function that has been divided by (x-k). The quotient q(x) is the result of the division, and the remainder r is the amount left over after the division.

Q: How do I determine the value of k?

A: The value of k is the root of the divisor (x-k). In other words, k is the value that makes the divisor equal to zero. For example, if the divisor is (x+1), then k = -1.

Q: What is the difference between the quotient and the remainder?

A: The quotient is the result of the division, and it is a polynomial function that represents the amount of the dividend that is divided by the divisor. The remainder is the amount left over after the division, and it is a constant or a polynomial function of a lower degree than the divisor.

Q: Can I use polynomial long division to divide a polynomial function by a linear factor?

A: Yes, you can use polynomial long division to divide a polynomial function by a linear factor. In fact, polynomial long division is a powerful tool for dividing polynomial functions by linear factors.

Q: What are some common applications of the form (x-k)q(x) + r?

A: The form (x-k)q(x) + r has many applications in mathematics and science. Some common applications include:

Polynomial division: The form (x-k)q(x) + r is used to divide polynomial functions by linear factors.
Factoring: The form (x-k)q(x) + r is used to factor polynomial functions into the product of two or more linear factors.
Graphing: The form (x-k)q(x) + r is used to graph polynomial functions by plotting the quotient and the remainder.

Q: Can I use the form (x-k)q(x) + r to divide a polynomial function by a quadratic factor?

A: Yes, you can use the form (x-k)q(x) + r to divide a polynomial function by a quadratic factor. However, the process is more complex and requires the use of polynomial long division or synthetic division.

Q: What are some common mistakes to avoid when using the form (x-k)q(x) + r?

A: Some common mistakes to avoid when using the form (x-k)q(x) + r include:

Not checking the degree of the divisor and the dividend.
Not using the correct value of k.
Not performing the division correctly.
Not checking the remainder for accuracy.

Conclusion

In this article, we have answered some frequently asked questions related to expressing a polynomial function in the form (x-k)q(x) + r. We have discussed the significance of the form, how to determine the value of k, and some common applications of the form. We have also highlighted some common mistakes to avoid when using the form. By following these guidelines, you can use the form (x-k)q(x) + r to divide polynomial functions by linear factors and to factor polynomial functions into the product of two or more linear factors.

Final Answer

The final answer is:

The form (x-k)q(x) + r is a powerful tool for dividing polynomial functions by linear factors and for factoring polynomial functions into the product of two or more linear factors. By following the guidelines outlined in this article, you can use the form to solve a wide range of mathematical problems.