Polynomial Division A0Cbfe

1. **State the problem:** Divide the polynomial $$5x^3 + x^2 + 18x$$ by $$x + 3$$ using long division. 2. **Recall the division algorithm for polynomials:** $$\text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder}$$ where the degree of the remainder is less than the degree of the divisor. 3. **Set up the long division:** Divide $$5x^3$$ (the leading term of the dividend) by $$x$$ (the leading term of the divisor): $$\frac{5x^3}{x} = 5x^2$$ This is the first term of the quotient. 4. **Multiply the divisor by this term:** $$(x + 3)(5x^2) = 5x^3 + 15x^2$$ 5. **Subtract this from the dividend:** $$\begin{aligned} & (5x^3 + x^2 + 18x) - (5x^3 + 15x^2) \\ &= 5x^3 - 5x^3 + x^2 - 15x^2 + 18x \\ &= 0 - 14x^2 + 18x = -14x^2 + 18x \end{aligned}$$ 6. **Repeat the process with the new polynomial $$-14x^2 + 18x$$:** Divide $$-14x^2$$ by $$x$$: $$\frac{-14x^2}{x} = -14x$$ Add $$-14x$$ to the quotient. 7. **Multiply the divisor by $$-14x$$:** $$(x + 3)(-14x) = -14x^2 - 42x$$ 8. **Subtract this from the current polynomial:** $$\begin{aligned} & (-14x^2 + 18x) - (-14x^2 - 42x) \\ &= -14x^2 + 18x + 14x^2 + 42x \\ &= 0 + 60x = 60x \end{aligned}$$ 9. **Repeat with $$60x$$:** Divide $$60x$$ by $$x$$: $$\frac{60x}{x} = 60$$ Add $$60$$ to the quotient. 10. **Multiply the divisor by $$60$$:** $$(x + 3)(60) = 60x + 180$$ 11. **Subtract this from the current polynomial:** $$\begin{aligned} & (60x) - (60x + 180) \\ &= 60x - 60x - 180 \\ &= -180 \end{aligned}$$ 12. **Since the remainder $$-180$$ is a constant and the divisor degree is 1, the remainder degree is less than divisor degree, so we stop.** 13. **Write the division algorithm:** $$5x^3 + x^2 + 18x = (x + 3)(5x^2 - 14x + 60) - 180$$ **Final answer:** Quotient: $$5x^2 - 14x + 60$$ Remainder: $$-180$$