What Is The Remainder When $\left(3x^3 - 2x^2 + 4x - 3\right$\] Is Divided By $\left(x^2 + 3x + 3\right$\]?A. 30 B. $3x - 11$ C. $28x - 36$ D. $28x + 30$

Introduction

In algebra, polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. The remainder is a polynomial of lesser degree than the divisor. In this article, we will explore the concept of polynomial division and find the remainder when a given polynomial is divided by another polynomial.

Polynomial Division

Polynomial division is a process of dividing 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 the degree of the remainder is less than the degree of the divisor.

The Remainder Theorem

The remainder theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). This theorem can be extended to polynomial division by a polynomial of degree 2 or more.

Finding the Remainder

To find the remainder when a polynomial is divided by another polynomial, we can use the following 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.

Example

Let's consider the polynomial division problem:

What is the remainder when $\left(3x^3 - 2x^2 + 4x - 3\right)$ is divided by $\left(x^2 + 3x + 3\right)$?

To find the remainder, we can use the steps outlined above.

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 $3x^3$ and the highest degree term of the divisor is $x^2$. We can divide $3x^3$ by $x^2$ to get $3x$.

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

We can multiply the entire divisor by $3x$ to get $3x(x^2 + 3x + 3) = 3x^3 + 9x^2 + 9x$. Subtracting this from the dividend, we get:

$\left(3x^3 - 2x^2 + 4x - 3\right) - \left(3x^3 + 9x^2 + 9x\right) = -11x^2 - 5x - 3$

Step 3: Repeat the process until the degree of the remainder is less than the degree of the divisor

We can repeat the process by dividing the highest degree term of the remainder by the highest degree term of the divisor. The highest degree term of the remainder is $-11x^2$ and the highest degree term of the divisor is $x^2$. We can divide $-11x^2$ by $x^2$ to get $-11$.

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

We can multiply the entire divisor by $-11$ to get $-11(x^2 + 3x + 3) = -11x^2 - 33x - 33$. Subtracting this from the remainder, we get:

$\left(-11x^2 - 5x - 3\right) - \left(-11x^2 - 33x - 33\right) = 28x + 30$

Conclusion

In this article, we explored the concept of polynomial division and found the remainder when a given polynomial is divided by another polynomial. We used the steps outlined above to find the remainder and obtained the result $28x + 30$.

Answer

The remainder when $\left(3x^3 - 2x^2 + 4x - 3\right)$ is divided by $\left(x^2 + 3x + 3\right)$ is $28x + 30$.

Final Answer

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Q&A

Q: What is polynomial division?

A: Polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. The remainder is a polynomial of lesser degree than the divisor.

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is equal to f(a). This theorem can be extended to polynomial division by a polynomial of degree 2 or more.

Q: How do I find the remainder when a polynomial is divided by another polynomial?

A: To find the remainder, you can use the following 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 difference between a quotient and a remainder in polynomial division?

A: The quotient is the result of dividing the dividend by the divisor, while the remainder is the polynomial that is left over after the division process is complete.

Q: Can I use the remainder theorem to find the remainder when a polynomial is divided by a polynomial of degree 2 or more?

A: Yes, you can use the remainder theorem to find the remainder when a polynomial is divided by a polynomial of degree 2 or more. However, you will need to use the extended remainder theorem, which states that if a polynomial f(x) is divided by a polynomial of degree n, then the remainder will be a polynomial of degree less than n.

Q: How do I know when to stop the division process?

A: You can stop the division process when the degree of the remainder is less than the degree of the divisor. This is because the remainder will be a polynomial of lesser degree than the divisor, and further division will not produce a new remainder.

Q: Can I use polynomial division to find the roots of a polynomial?

A: Yes, you can use polynomial division to find the roots of a polynomial. If you divide a polynomial by (x - a), and the remainder is zero, then a is a root of the polynomial.

Q: What are some common applications of polynomial division?

A: Polynomial division has many applications in mathematics and science, including:

• Finding the roots of a polynomial
• Factoring a polynomial
• Solving systems of equations
• Finding the maximum or minimum of a function

Conclusion

In this article, we explored the concept of polynomial division and answered some common questions about the process. We also discussed the remainder theorem and its applications in mathematics and science.

Final Answer

The final answer is $28x + 30$.