find the quotient and remainder for $$ \frac{\left(5x^4-3x^2+6x-3\right)}{x+3} $$

To find the quotient and remainder of the division $ \frac{5x^4 - 3x^2 + 6x - 3}{x + 3} $, we will use polynomial long division. Here's the process step-by-step:

1. **Set up the division:** Write $ 5x^4 - 3x^2 + 6x - 3 $ as the dividend and $ x + 3 $ as the divisor.

\[
\begin{array}{r|rrrr}
x + 3 & 5x^4 & 0x^3 & -3x^2 & 6x & -3 \\
\end{array}
\]

2. **Divide the first term:** Divide the leading term of the dividend by the leading term of the divisor.

\[
\frac{5x^4}{x} = 5x^3
\]

Write $ 5x^3 $ as the first term of the quotient.

3. **Multiply and subtract:** Multiply $ 5x^3 $ by $ x + 3 $ and subtract from the dividend.

\[
(5x^4 + 15x^3) \rightarrow (5x^4 + 0x^3 - 3x^2 + 6x - 3) - (5x^4 + 15x^3) = -15x^3 - 3x^2 + 6x - 3
\]

4. **Repeat the process:** Repeat the process for $ -15x^3 $.

\[
\frac{-15x^3}{x} = -15x^2
\]

Write $ -15x^2 $ as the next term of the quotient.

\[
(-15x^3 - 45x^2) \rightarrow (-15x^3 - 3x^2 + 6x - 3) - (-15x^3 - 45x^2) = 42x^2 + 6x - 3
\]

5. **Continue the process:**

\[
\frac{42x^2}{x} = 42x
\]

Write $ 42x $ as the next term of the quotient.

\[
(42x^2 + 126x) \rightarrow (42x^2 + 6x - 3) - (42x^2 + 126x) = -120x - 3
\]

6. **One more time:**

\[
\frac{-120x}{x} = -120
\]

Write $ -120 $ as the next term of the quotient.

\[
(-120x - 360) \rightarrow (-120x - 3) - (-120x - 360) = 357
\]

7. **Remainder:**

After the last step, the remaining term is $ 357 $, which is the remainder.

8. **Conclusion:**

The quotient is $ 5x^3 - 15x^2 + 42x - 120 $ and the remainder is $ 357 $.

Therefore, the result of the division is:

\[
\frac{5x^4 - 3x^2 + 6x - 3}{x + 3} = 5x^3 - 15x^2 + 42x - 120 \quad \text{with a remainder of } 357.
\]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How can you verify the quotient and remainder of a polynomial division?
2. What is synthetic division, and how does it differ from polynomial long division?
3. Can you use polynomial long division for dividing higher-degree polynomials?
4. How does the Remainder Theorem apply to polynomial division?
5. What are the practical applications of polynomial division in calculus and algebra?

**Tip:** When performing polynomial division, always arrange the polynomial terms in descending order of their degrees and include any missing terms with a coefficient of zero.

Learn how to find the quotient and remainder when dividing the polynomial 5x^4 - 3x^2 + 6x - 3 by x + 3 using polynomial long division. This detailed guide covers key algebraic concepts and the Remainder Theorem. Suitable for high school students (Grades 10-12).

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