Divide the leading term of the dividend by the leading term of the divisor

Oct 13, 2025 by magicalceylontours.com 95 views

Introduction

Polynomial division is a fundamental concept in algebra that allows us to divide a polynomial by another polynomial. This process is essential in finding the zeros of a polynomial function, which are the values of x that make the function equal to zero. In this article, we will explore how to use polynomial division to find the zeros of two given polynomial functions.

Polynomial Division: A Step-by-Step Guide

Before we dive into the problems, let's review the steps involved in polynomial division.

Divide the leading term of the dividend by the leading term of the divisor: This will give us the first term of the quotient.
Multiply the entire divisor by the first term of the quotient: This will give us the first term of the product.
Subtract the product from the dividend: This will give us a new dividend.
Repeat steps 1-3 until we have no more terms to divide: This will give us the final quotient.

Problem 10: Finding Zeros of $f(x)=x^3-2x^2-21x-18$

Given the polynomial function $f(x)=x^3-2x^2-21x-18$ , we are asked to find all zeros of the function by first using polynomial division. We are also given that $x=-1$ is a zero of the function.

Step 1: Perform Polynomial Division

To find the zeros of the function, we need to perform polynomial division. We will divide the polynomial $f(x)$ by $x+1$ .

x^3 - 2x^2 - 21x - 18
-------------------------
x + 1

Step 2: Divide the Leading Term of the Dividend by the Leading Term of the Divisor

The leading term of the dividend is $x^3$ , and the leading term of the divisor is $x$ . Therefore, the first term of the quotient is $x^2$ .

x^2 - 2x - 21
-------------------------
x + 1

Step 3: Multiply the Entire Divisor by the First Term of the Quotient

We multiply the entire divisor $x+1$ by the first term of the quotient $x^2$ . This gives us $x^3+x^2$ .

x^3 - 2x^2 - 21x - 18
-------------------------
x^3 + x^2

Step 4: Subtract the Product from the Dividend

We subtract the product $x^3+x^2$ from the dividend $x^3-2x^2-21x-18$ . This gives us $-3x^2-21x-18$ .

-3x^2 - 21x - 18
-------------------------
x + 1

Step 5: Repeat Steps 1-4

We repeat the process until we have no more terms to divide. The final quotient is $x^2-3x-18$ .

x^2 - 3x - 18
-------------------------
x + 1

Step 6: Factor the Quotient

We can factor the quotient $x^2-3x-18$ as $(x-6)(x+3)$ .

(x - 6)(x + 3)
-------------------------
x + 1

Step 7: Find the Zeros of the Function

We can now find the zeros of the function by setting the quotient equal to zero. We have $(x-6)(x+3)=0$ , which gives us $x=6$ or $x=-3$ .

Problem 12: Finding Zeros of $h(x)=x^5+4x^4-19x^3-82x^2+48x+288$

Given the polynomial function $h(x)=x^5+4x^4-19x^3-82x^2+48x+288$ , we are asked to find all zeros of the function by first using polynomial division. We are also given that $x=1$ is a zero of the function.

Step 1: Perform Polynomial Division

To find the zeros of the function, we need to perform polynomial division. We will divide the polynomial $h(x)$ by $x-1$ .

x^5 + 4x^4 - 19x^3 - 82x^2 + 48x + 288
-------------------------
x - 1

Step 2: Divide the Leading Term of the Dividend by the Leading Term of the Divisor

The leading term of the dividend is $x^5$ , and the leading term of the divisor is $x$ . Therefore, the first term of the quotient is $x^4$ .

x^4 + 5x^3 - 14x^2 - 71x - 288
-------------------------
x - 1

Step 3: Multiply the Entire Divisor by the First Term of the Quotient

We multiply the entire divisor $x-1$ by the first term of the quotient $x^4$ . This gives us $x^5-x^4$ .

x^5 + 4x^4 - 19x^3 - 82x^2 + 48x + 288
-------------------------
x^5 - x^4

Step 4: Subtract the Product from the Dividend

We subtract the product $x^5-x^4$ from the dividend $x^5+4x^4-19x^3-82x^2+48x+288$ . This gives us $5x^4-19x^3-82x^2+48x+288$ .

5x^4 - 19x^3 - 82x^2 + 48x + 288
-------------------------
x - 1

Step 5: Repeat Steps 1-4

We repeat the process until we have no more terms to divide. The final quotient is $5x^4-19x^3-82x^2+48x+288$ .

5x^4 - 19x^3 - 82x^2 + 48x + 288
-------------------------
x - 1

Step 6: Factor the Quotient

We can factor the quotient $5x^4-19x^3-82x^2+48x+288$ as $(x-1)(5x^3-14x^2-68x-288)$ .

(x - 1)(5x^3 - 14x^2 - 68x - 288)
-------------------------
x - 1

Step 7: Find the Zeros of the Function

We can now find the zeros of the function by setting the quotient equal to zero. We have $(x-1)(5x^3-14x^2-68x-288)=0$ , which gives us $x=1$ or $5x^3-14x^2-68x-288=0$ .

Conclusion

In this article, we have used polynomial division to find the zeros of two given polynomial functions. We have shown that polynomial division is a powerful tool for finding the zeros of polynomial functions. By following the steps outlined in this article, we can find the zeros of any polynomial function.

References

Introduction

In our previous article, we explored how to use polynomial division to find the zeros of two given polynomial functions. In this article, we will answer some common questions related to polynomial division and provide additional examples to help you practice.

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial. It is a fundamental concept in algebra that allows us to divide a polynomial by another polynomial and find the quotient and remainder.

Q: Why do we need to perform polynomial division?

A: We need to perform polynomial division to find the zeros of a polynomial function. By dividing the polynomial by a linear factor, we can find the zeros of the function.

Q: How do I perform polynomial division?

A: To perform polynomial division, follow these steps:

Divide the leading term of the dividend by the leading term of the divisor: This will give us the first term of the quotient.
Multiply the entire divisor by the first term of the quotient: This will give us the first term of the product.
Subtract the product from the dividend: This will give us a new dividend.
Repeat steps 1-3 until we have no more terms to divide: This will give us the final quotient.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the amount left over after we have divided the dividend by the divisor. It is the term that is not divisible by the divisor.

Q: How do I find the zeros of a polynomial function using polynomial division?

A: To find the zeros of a polynomial function using polynomial division, follow these steps:

Perform polynomial division: Divide the polynomial by a linear factor.
Find the quotient: The quotient is the result of the division.
Set the quotient equal to zero: Set the quotient equal to zero to find the zeros of the function.
Solve for x: Solve for x to find the zeros of the function.

Q: What are some common mistakes to avoid when performing polynomial division?

A: Some common mistakes to avoid when performing polynomial division include:

Not following the order of operations: Make sure to follow the order of operations when performing polynomial division.
Not multiplying the entire divisor by the first term of the quotient: Make sure to multiply the entire divisor by the first term of the quotient.
Not subtracting the product from the dividend: Make sure to subtract the product from the dividend.

Q: How can I practice polynomial division?

A: You can practice polynomial division by using online resources, such as Khan Academy or Mathway, or by working through practice problems in a textbook.

Q: What are some real-world applications of polynomial division?

A: Polynomial division has many real-world applications, including:

Engineering: Polynomial division is used in engineering to design and analyze systems.
Computer Science: Polynomial division is used in computer science to solve problems related to algorithms and data structures.
Economics: Polynomial division is used in economics to model and analyze economic systems.

Conclusion

In this article, we have answered some common questions related to polynomial division and provided additional examples to help you practice. Polynomial division is a powerful tool for finding the zeros of polynomial functions, and it has many real-world applications.