Divide The Following Polynomials, Then Place The Answer In The Proper Location On The Grid. Write The Answer In Descending Powers Of X X X . ( 4 X 3 + 3 X 2 − 30 X − 10 ) ÷ ( X − 3 \left(4x^3 + 3x^2 - 30x - 10\right) \div (x - 3 ( 4 X 3 + 3 X 2 − 30 X − 10 ) ÷ ( X − 3 ]

Introduction

In algebra, polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will focus on dividing the polynomial $4x^3 + 3x^2 - 30x - 10$ by $x - 3$ and placing the answer in the proper location on the grid.

Understanding Polynomial Division

Polynomial division is similar to long division, where we divide a polynomial by another polynomial to obtain a quotient and a remainder. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until we obtain a remainder that is less than the degree of the divisor.

Setting Up the Division

To divide the polynomial $4x^3 + 3x^2 - 30x - 10$ by $x - 3$, we need to set up the division in the following format:

  ____________________
x - 3 | 4x^3 + 3x^2 - 30x - 10

Performing the Division

To perform the division, we need to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, we divide $4x^3$ by $x$, which gives us $4x^2$. We then multiply the entire divisor by $4x^2$ and subtract it from the dividend.

  ____________________
x - 3 | 4x^3 + 3x^2 - 30x - 10
  - (4x^3 - 12x^2)
  ____________________
  15x^2 - 30x - 10

Continuing the Division

We then repeat the process by dividing the highest degree term of the new dividend by the highest degree term of the divisor. In this case, we divide $15x^2$ by $x$, which gives us $15x$. We then multiply the entire divisor by $15x$ and subtract it from the new dividend.

  ____________________
x - 3 | 4x^3 + 3x^2 - 30x - 10
  - (4x^3 - 12x^2)
  ____________________
  15x^2 - 30x - 10
  - (15x^2 - 45x)
  ____________________
  -15x - 10

Finalizing the Division

We then repeat the process one more time by dividing the highest degree term of the new dividend by the highest degree term of the divisor. In this case, we divide $-15x$ by $x$, which gives us $-15$. We then multiply the entire divisor by $-15$ and subtract it from the new dividend.

  ____________________
x - 3 | 4x^3 + 3x^2 - 30x - 10
  - (4x^3 - 12x^2)
  ____________________
  15x^2 - 30x - 10
  - (15x^2 - 45x)
  ____________________
  -15x - 10
  - (-15x + 45)
  ____________________
  35

Writing the Answer in Descending Powers of $x$

The final answer is $4x^2 + 15x - 15$ with a remainder of $35$. We can write the answer in descending powers of $x$ as follows:

$$\frac{4x^3 + 3x^2 - 30x - 10}{x - 3} = 4x^2 + 15x - 15 + \frac{35}{x - 3}$$

Conclusion

In this article, we have divided the polynomial $4x^3 + 3x^2 - 30x - 10$ by $x - 3$ and placed the answer in the proper location on the grid. We have also written the answer in descending powers of $x$. Polynomial division is an essential process in algebra, and it is used to solve equations, find roots, and simplify expressions. By following the steps outlined in this article, you can perform polynomial division and write the answer in descending powers of $x$.

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. In our previous article, we discussed how to divide the polynomial $4x^3 + 3x^2 - 30x - 10$ by $x - 3$ and placed the answer in the proper location on the grid. In this article, we will answer some frequently asked questions about polynomial division.

Q: What is polynomial division?

A: Polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. It is similar to long division, where we divide a polynomial by another polynomial to obtain a quotient and a remainder.

Q: Why is polynomial division important?

A: Polynomial division is important because it is used to solve equations, find roots, and simplify expressions. It is also used in various fields such as engineering, physics, and computer science.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to follow these steps:

1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
2. Multiply the entire divisor by the result and subtract it from the dividend.
3. Repeat the process until you obtain a remainder that is less than the degree of the divisor.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the amount left over after dividing the dividend by the divisor. It is usually a polynomial of a lower degree than the divisor.

Q: Can the remainder be zero?

A: Yes, the remainder can be zero. This occurs when the dividend is exactly divisible by the divisor.

Q: How do I write the answer in descending powers of $x$?

A: To write the answer in descending powers of $x$, you need to arrange the terms of the quotient in descending order of their powers.

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

A: Polynomial division and long division are similar, but polynomial division involves dividing polynomials, while long division involves dividing integers.

Q: Can I use polynomial division to divide a polynomial by a binomial?

A: Yes, you can use polynomial division to divide a polynomial by a binomial. In fact, this is one of the most common applications of polynomial division.

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 divisor and add the remainder. If the result is equal to the dividend, 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 following the correct order of operations
• Not multiplying the entire divisor by the result
• Not subtracting the result from the dividend
• Not checking the work

Conclusion

In this article, we have answered some frequently asked questions about polynomial division. Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another to obtain a quotient and a remainder. By following the steps outlined in this article, you can perform polynomial division and write the answer in descending powers of $x$.