Divide. If The Polynomial Does Not Divide Evenly, Include The Remainder As A Fraction. ${ (-40s^4 - 102s^3 + 54s - 68) \div (4s + 7) }$

Introduction

Polynomial Division is a mathematical operation that involves dividing one polynomial by another. It is a crucial concept in algebra and is used to simplify complex expressions and solve equations. In this article, we will focus on dividing a polynomial by a linear expression, and we will use the given example to illustrate the process.

The Problem

We are given the polynomial $(-40s^4 - 102s^3 + 54s - 68)$ and we need to divide it by $(4s + 7)$. Our goal is to find the quotient and the remainder, if any.

Step 1: Set Up the Division

To divide the polynomial by the linear expression, we need to set up the division in the following format:

$$\frac{(-40s^4 - 102s^3 + 54s - 68)}{(4s + 7)}$$

Step 2: Divide the Leading Term

We start by dividing the leading term of the polynomial, which is $-40s^4$, by the leading term of the linear expression, which is $4s$. This gives us:

$$\frac{-40s^4}{4s} = -10s^3$$

Step 3: Multiply and Subtract

Next, we multiply the entire linear expression by the quotient we obtained in the previous step, which is $-10s^3$. This gives us:

$$-10s^3(4s + 7) = -40s^4 - 70s^3$$

We then subtract this result from the original polynomial:

$$(-40s^4 - 102s^3 + 54s - 68) - (-40s^4 - 70s^3) = -32s^3 + 54s - 68$$

Step 4: Repeat the Process

We now repeat the process by dividing the leading term of the resulting polynomial, which is $-32s^3$, by the leading term of the linear expression, which is $4s$. This gives us:

$$\frac{-32s^3}{4s} = -8s^2$$

Step 5: Multiply and Subtract Again

Next, we multiply the entire linear expression by the quotient we obtained in the previous step, which is $-8s^2$. This gives us:

$$-8s^2(4s + 7) = -32s^3 - 56s^2$$

We then subtract this result from the previous polynomial:

$$(-32s^3 + 54s - 68) - (-32s^3 - 56s^2) = 56s^2 + 54s - 68$$

Step 6: Repeat the Process Again

We now repeat the process by dividing the leading term of the resulting polynomial, which is $56s^2$, by the leading term of the linear expression, which is $4s$. This gives us:

$$\frac{56s^2}{4s} = 14s$$

Step 7: Multiply and Subtract Again

Next, we multiply the entire linear expression by the quotient we obtained in the previous step, which is $14s$. This gives us:

$$14s(4s + 7) = 56s^2 + 98s$$

We then subtract this result from the previous polynomial:

$$(56s^2 + 54s - 68) - (56s^2 + 98s) = -44s - 68$$

Step 8: Repeat the Process Again

We now repeat the process by dividing the leading term of the resulting polynomial, which is $-44s$, by the leading term of the linear expression, which is $4s$. This gives us:

$$\frac{-44s}{4s} = -11$$

Step 9: Multiply and Subtract Again

Next, we multiply the entire linear expression by the quotient we obtained in the previous step, which is $-11$. This gives us:

$$-11(4s + 7) = -44s - 77$$

We then subtract this result from the previous polynomial:

$$(-44s - 68) - (-44s - 77) = 9$$

Conclusion

We have now completed the polynomial division. The quotient is $-10s^3 - 8s^2 + 14s - 11$ and the remainder is $9$. Therefore, we can write the result of the division as:

$$\frac{(-40s^4 - 102s^3 + 54s - 68)}{(4s + 7)} = -10s^3 - 8s^2 + 14s - 11 + \frac{9}{4s + 7}$$

This result shows that the polynomial $(-40s^4 - 102s^3 + 54s - 68)$ can be divided by $(4s + 7)$ to give a quotient of $-10s^3 - 8s^2 + 14s - 11$ and a remainder of $9$.

Final Answer

The final answer is $\boxed{-10s^3 - 8s^2 + 14s - 11 + \frac{9}{4s + 7}}$.

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation that helps us simplify complex expressions and solve equations. In this article, we will address some of the most frequently asked questions about polynomial division.

Q: What is polynomial division?

A: Polynomial division is a mathematical operation that involves dividing one polynomial by another. It is a process of finding the quotient and remainder when one polynomial is divided by another.

Q: How do I divide a polynomial by a linear expression?

A: To divide a polynomial by a linear expression, you need to follow these steps:

1. Set up the division in the correct format.
2. Divide the leading term of the polynomial by the leading term of the linear expression.
3. Multiply the entire linear expression by the quotient obtained in step 2.
4. Subtract the result from step 3 from the original polynomial.
5. Repeat the process until you obtain a remainder that is less than the degree of the linear expression.

Q: What is the quotient and remainder in polynomial division?

A: The quotient is the result of dividing the polynomial by the linear expression, while the remainder is the amount left over after the division.

Q: How do I handle a remainder in polynomial division?

A: If the remainder is not zero, you can write the result of the division as the quotient plus the remainder divided by the linear expression.

Q: Can I divide a polynomial by a polynomial of higher degree?

A: No, you cannot divide a polynomial by a polynomial of higher degree. This is because the degree of the quotient will be less than the degree of the dividend, which is not possible.

Q: What is the difference between polynomial division and long division?

A: Polynomial division and long division are both methods of dividing numbers, but they are used for different types of numbers. Polynomial division is used for dividing polynomials, while long division is used for dividing integers.

Q: Can I use a calculator to perform polynomial division?

A: Yes, you can use a calculator to perform polynomial division. Many calculators have a built-in function for polynomial division that can simplify the process.

Q: How do I check my work in polynomial division?

A: To check your work in polynomial division, you can multiply the quotient by the linear expression and add the remainder. If the result is equal to the original polynomial, then your work is correct.

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

A: Some common mistakes to avoid in polynomial division include:

• Not setting up the division correctly
• Not following the correct order of operations
• Not handling the remainder correctly
• Not checking your work

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation that helps us simplify complex expressions and solve equations. By following the steps outlined in this article, you can perform polynomial division with confidence and accuracy.

Final Answer

The final answer is $\boxed{0}$, but the real answer is the knowledge and understanding of polynomial division that you have gained from this article.