Poly Division Cc70F9

1. **State the problem:** We are given two polynomials $$f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$$ and $$p(x) = x + 4$$. We want to perform polynomial division to find the quotient and remainder when $$f(x)$$ is divided by $$p(x)$$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by that result, subtract, and repeat with the remainder until the degree of the remainder is less than the divisor. 3. **Step 1:** Divide the leading term of $$f(x)$$, which is $$2x^5$$, by the leading term of $$p(x)$$, which is $$x$$: $$\frac{2x^5}{x} = 2x^4$$ 4. **Step 2:** Multiply $$p(x)$$ by $$2x^4$$: $$2x^4(x + 4) = 2x^5 + 8x^4$$ 5. **Step 3:** Subtract this from $$f(x)$$: $$\left(2x^5 - 3x^3 + 2x^2 + x - 3\right) - \left(2x^5 + 8x^4\right) = -8x^4 - 3x^3 + 2x^2 + x - 3$$ 6. **Step 4:** Divide the leading term of the new polynomial $$-8x^4$$ by $$x$$: $$\frac{-8x^4}{x} = -8x^3$$ 7. **Step 5:** Multiply $$p(x)$$ by $$-8x^3$$: $$-8x^3(x + 4) = -8x^4 - 32x^3$$ 8. **Step 6:** Subtract: $$\left(-8x^4 - 3x^3 + 2x^2 + x - 3\right) - \left(-8x^4 - 32x^3\right) = 29x^3 + 2x^2 + x - 3$$ 9. **Step 7:** Divide $$29x^3$$ by $$x$$: $$\frac{29x^3}{x} = 29x^2$$ 10. **Step 8:** Multiply $$p(x)$$ by $$29x^2$$: $$29x^2(x + 4) = 29x^3 + 116x^2$$ 11. **Step 9:** Subtract: $$\left(29x^3 + 2x^2 + x - 3\right) - \left(29x^3 + 116x^2\right) = -114x^2 + x - 3$$ 12. **Step 10:** Divide $$-114x^2$$ by $$x$$: $$\frac{-114x^2}{x} = -114x$$ 13. **Step 11:** Multiply $$p(x)$$ by $$-114x$$: $$-114x(x + 4) = -114x^2 - 456x$$ 14. **Step 12:** Subtract: $$\left(-114x^2 + x - 3\right) - \left(-114x^2 - 456x\right) = 457x - 3$$ 15. **Step 13:** Divide $$457x$$ by $$x$$: $$\frac{457x}{x} = 457$$ 16. **Step 14:** Multiply $$p(x)$$ by $$457$$: $$457(x + 4) = 457x + 1828$$ 17. **Step 15:** Subtract: $$\left(457x - 3\right) - \left(457x + 1828\right) = -1831$$ 18. **Conclusion:** The quotient is $$2x^4 - 8x^3 + 29x^2 - 114x + 457$$ and the remainder is $$-1831$$. **Final answer:** $$\frac{f(x)}{p(x)} = 2x^4 - 8x^3 + 29x^2 - 114x + 457 + \frac{-1831}{x + 4}$$