Question: synthetic division of $x^4+2x^3-4x^2-10x+5$ by $x^2-5$
1. **State the problem:** We want to divide the polynomial $$x^4+2x^3-4x^2-10x+5$$ by the polynomial $$x^2-5$$ using synthetic division.
2. **Important note:** Synthetic division is typically used for dividing by linear polynomials of the form $$x - c$$. Since $$x^2 - 5$$ is quadratic, we cannot use the standard synthetic division method directly. Instead, we use polynomial long division or a modified synthetic division method.
3. **Set up polynomial long division:** Divide $$x^4+2x^3-4x^2-10x+5$$ by $$x^2-5$$.
4. **Divide the leading terms:** $$\frac{x^4}{x^2} = x^2$$.
5. **Multiply divisor by $$x^2$$:** $$x^2(x^2 - 5) = x^4 - 5x^2$$.
6. **Subtract:**
$$
(x^4 + 2x^3 - 4x^2 - 10x + 5) - (x^4 - 5x^2) = 2x^3 + ( -4x^2 + 5x^2) - 10x + 5 = 2x^3 + x^2 - 10x + 5
$$
7. **Divide leading terms again:** $$\frac{2x^3}{x^2} = 2x$$.
8. **Multiply divisor by $$2x$$:** $$2x(x^2 - 5) = 2x^3 - 10x$$.
9. **Subtract:**
$$
(2x^3 + x^2 - 10x + 5) - (2x^3 - 10x) = x^2 + ( -10x + 10x) + 5 = x^2 + 5
$$
10. **Divide leading terms:** $$\frac{x^2}{x^2} = 1$$.
11. **Multiply divisor by $$1$$:** $$1(x^2 - 5) = x^2 - 5$$.
12. **Subtract:**
$$
(x^2 + 5) - (x^2 - 5) = 0 + 10 = 10
$$
13. **Result:** The quotient is $$x^2 + 2x + 1$$ and the remainder is $$10$$.
14. **Final answer:**
$$
\frac{x^4+2x^3-4x^2-10x+5}{x^2-5} = x^2 + 2x + 1 + \frac{10}{x^2 - 5}
$$