Compute \left(x^4-5 X^3+4 X^2-15 X+3\right) \div\left(x^2+3\right ].A. X 2 − 8 X + 1 X^2-8 X+1 X 2 − 8 X + 1 B. X 2 − 8 X − 20 + − 75 X + 3 X 2 + 3 X^2-8 X-20+\frac{-75 X+3}{x^2+3} X 2 − 8 X − 20 + X 2 + 3 − 75 X + 3 ​ C. X 2 − 5 X + 1 X^2-5 X+1 X 2 − 5 X + 1 D. X 2 − 5 X + 7 + − 30 X + 24 X 2 + 3 X^2-5 X+7+\frac{-30 X+24}{x^2+3} X 2 − 5 X + 7 + X 2 + 3 − 30 X + 24 ​

**Dividing Polynomials: A Step-by-Step Guide** =====================================================

What is Polynomial Division?

Polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. It is an essential concept in algebra and is used to simplify complex expressions and solve equations.

Why is Polynomial Division Important?

Polynomial division is used in various fields such as engineering, physics, and computer science. It is used to model real-world problems, solve equations, and optimize systems. In addition, polynomial division is used in cryptography and coding theory to develop secure algorithms.

How to Divide Polynomials?

To divide polynomials, we follow these steps:

1. Write the dividend and divisor: Write the dividend (the polynomial being divided) and the divisor (the polynomial by which we are dividing) in the correct order.
2. Determine the degree of the divisor: Determine the degree of the divisor, which is the highest power of the variable in the divisor.
3. Divide the leading term of the dividend: Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient.
4. Multiply the divisor by the quotient term: Multiply the divisor by the quotient term obtained in step 3.
5. Subtract the product from the dividend: Subtract the product obtained in step 4 from the dividend.
6. Repeat steps 3-5: Repeat steps 3-5 until the degree of the remaining dividend is less than the degree of the divisor.
7. Write the final quotient and remainder: Write the final quotient and remainder.

Example: Dividing Polynomials

Let's consider the example of dividing the polynomial $x^4-5x^3+4x^2-15x+3$ by $x^2+3$.

Step 1: Write the dividend and divisor

The dividend is $x^4-5x^3+4x^2-15x+3$ and the divisor is $x^2+3$.

Step 2: Determine the degree of the divisor

The degree of the divisor is 2, which is the highest power of the variable in the divisor.

Step 3: Divide the leading term of the dividend

The leading term of the dividend is $x^4$ and the leading term of the divisor is $x^2$. Dividing $x^4$ by $x^2$ gives $x^2$.

Step 4: Multiply the divisor by the quotient term

Multiplying the divisor $x^2+3$ by the quotient term $x^2$ gives $x^4+3x^2$.

Step 5: Subtract the product from the dividend

Subtracting $x^4+3x^2$ from the dividend $x^4-5x^3+4x^2-15x+3$ gives $-5x^3-x^2-15x+3$.

Step 6: Repeat steps 3-5

Repeating steps 3-5, we get:

• $-5x^3-x^2-15x+3$ divided by $x^2+3$ gives $-5x+1$.
• $-5x+1$ multiplied by $x^2+3$ gives $-5x^3-12x+3$.
• Subtracting $-5x^3-12x+3$ from $-5x^3-x^2-15x+3$ gives $-x^2-3x$.

Step 7: Write the final quotient and remainder

The final quotient is $x^2-5x+1$ and the remainder is $-x^2-3x$.

Answer: The final answer is $x^2-5x+1+\frac{-x^2-3x}{x^2+3}$.

Q&A

Q: What is the degree of the divisor in polynomial division? A: The degree of the divisor is the highest power of the variable in the divisor.

Q: How do I determine the first term of the quotient in polynomial division? A: The first term of the quotient is obtained by dividing the leading term of the dividend by the leading term of the divisor.

Q: What is the remainder in polynomial division? A: The remainder is the expression that is left after the division process is complete.

Q: How do I write the final quotient and remainder in polynomial division? A: The final quotient and remainder are written in the form of a quotient plus a remainder divided by the divisor.

Q: What is the purpose of polynomial division? A: The purpose of polynomial division is to simplify complex expressions and solve equations.

Q: How is polynomial division used in real-world applications? A: Polynomial division is used in various fields such as engineering, physics, and computer science to model real-world problems, solve equations, and optimize systems.

Q: What is the difference between polynomial division and long division? A: Polynomial division is used to divide polynomials, while long division is used to divide integers.