Divide $32x^3 + 48x^2 - 40x$ By $8x$.A. $4x^2 - 6x + 5$ B. \$4x^2 + 6x - 5$[/tex\] C. $4x^3 - 6x^2 + 5$ D. $4x^3 + 6x^2 - 5$

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Introduction

In this article, we will be performing polynomial division, which is a fundamental concept in algebra. Polynomial division is used to divide a polynomial by another polynomial, and it is an essential tool in mathematics, particularly in calculus and algebra. In this discussion, we will be dividing the polynomial $32x^3 + 48x^2 - 40x$ by $8x$.

Understanding Polynomial Division

Polynomial division is similar to long division, but it is used for polynomials instead of integers. 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 the degree of the remainder is less than the degree of the divisor.

Dividing $32x^3 + 48x^2 - 40x$ by $8x$

To divide $32x^3 + 48x^2 - 40x$ by $8x$, we will follow the steps of polynomial division.

Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor

The highest degree term of the dividend is $32x^3$, and the highest degree term of the divisor is $8x$. Dividing $32x^3$ by $8x$ gives us $4x^2$.

Step 2: Multiply the entire divisor by the result and subtract it from the dividend

Multiplying the entire divisor $8x$ by $4x^2$ gives us $32x^3$. Subtracting this from the dividend $32x^3 + 48x^2 - 40x$ gives us $48x^2 - 40x - 32x^3$.

Step 3: Repeat the process

The highest degree term of the new dividend is $48x^2$, and the highest degree term of the divisor is $8x$. Dividing $48x^2$ by $8x$ gives us $6x$.

Step 4: Multiply the entire divisor by the result and subtract it from the dividend

Multiplying the entire divisor $8x$ by $6x$ gives us $48x^2$. Subtracting this from the new dividend $48x^2 - 40x - 32x^3$ gives us $-40x + 32x^3$.

Step 5: Repeat the process

The highest degree term of the new dividend is $-40x$, and the highest degree term of the divisor is $8x$. Dividing $-40x$ by $8x$ gives us $-5$.

Step 6: Multiply the entire divisor by the result and subtract it from the dividend

Multiplying the entire divisor $8x$ by $-5$ gives us $-40x$. Subtracting this from the new dividend $-40x + 32x^3$ gives us $32x^3$.

Conclusion

After performing the polynomial division, we get the quotient $4x^2 - 6x + 5$ and the remainder $0$. Therefore, the correct answer is:

A. $4x^2 - 6x + 5$

This is the final answer to the problem.

Introduction

In our previous article, we performed polynomial division to divide the polynomial $32x^3 + 48x^2 - 40x$ by $8x$. In this article, we will answer some frequently asked questions related to polynomial division and provide additional examples to help you understand the concept better.

Q&A

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial. It is similar to long division, but it is used for polynomials instead of integers.

Q: Why do we need polynomial division?

A: Polynomial division is used to simplify complex polynomials and to find the quotient and remainder when a polynomial is divided by another polynomial.

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 the degree of the remainder is less than the degree of the divisor.

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

A: The quotient is the result of the division, and the remainder is the amount left over after the division.

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

A: Yes, you can use polynomial division to divide a polynomial by a constant. In this case, the constant is considered as a polynomial with a degree of 0.

Q: Can I use polynomial division to divide a polynomial by a polynomial with a degree of 0?

A: No, you cannot use polynomial division to divide a polynomial by a polynomial with a degree of 0. In this case, the result will be the original polynomial.

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

A: Polynomial division is similar to long division, but it is used for polynomials instead of integers. The main difference is that polynomial division involves dividing polynomials, while long division involves dividing integers.

Examples

Example 1: Divide $2x^3 + 3x^2 - 4x$ by $x$

To divide $2x^3 + 3x^2 - 4x$ by $x$, we follow the steps of polynomial division.

  1. Divide the highest degree term of the dividend by the highest degree term of the divisor: $2x^3 \div x = 2x^2$
  2. Multiply the entire divisor by the result and subtract it from the dividend: $2x^2 \cdot x = 2x^3$, subtracting this from the dividend gives us $3x^2 - 4x - 2x^3$
  3. Repeat the process: $3x^2 \div x = 3x$
  4. Multiply the entire divisor by the result and subtract it from the dividend: $3x \cdot x = 3x^2$, subtracting this from the dividend gives us $-4x - 3x^2$
  5. Repeat the process: $-4x \div x = -4$
  6. Multiply the entire divisor by the result and subtract it from the dividend: $-4 \cdot x = -4x$, subtracting this from the dividend gives us $-3x^2$

The quotient is $2x^2 + 3x - 4$ and the remainder is $-3x^2$.

Example 2: Divide $x^3 + 2x^2 - 3x$ by $x^2$

To divide $x^3 + 2x^2 - 3x$ by $x^2$, we follow the steps of polynomial division.

  1. Divide the highest degree term of the dividend by the highest degree term of the divisor: $x^3 \div x^2 = x$
  2. Multiply the entire divisor by the result and subtract it from the dividend: $x \cdot x^2 = x^3$, subtracting this from the dividend gives us $2x^2 - 3x - x^3$
  3. Repeat the process: $2x^2 \div x^2 = 2$
  4. Multiply the entire divisor by the result and subtract it from the dividend: $2 \cdot x^2 = 2x^2$, subtracting this from the dividend gives us $-3x - 2x^2$
  5. Repeat the process: $-3x \div x^2 = -3x^{-1}$
  6. Multiply the entire divisor by the result and subtract it from the dividend: $-3x^{-1} \cdot x^2 = -3x$, subtracting this from the dividend gives us $-2x^2$

The quotient is $x + 2$ and the remainder is $-2x^2 - 3x$.

Conclusion

In this article, we answered some frequently asked questions related to polynomial division and provided additional examples to help you understand the concept better. Polynomial division is a powerful tool in mathematics that allows us to simplify complex polynomials and to find the quotient and remainder when a polynomial is divided by another polynomial.