Factor Check 1C9E50

1. **Problem Statement:** Determine if the given polynomials are divisible by the given factors by finding the remainder when divided. 2. **Factor Theorem:** A polynomial $f(x)$ is divisible by $(x - a)$ if and only if $f(a) = 0$. The remainder when dividing by $(x - a)$ is $f(a)$. --- ### Part 1: Is $x + 2$ a factor of $x^3 + 3x^2 + 5x + 6$? 3. Rewrite factor as $x - (-2)$, so $a = -2$. 4. Evaluate $f(-2) = (-2)^3 + 3(-2)^2 + 5(-2) + 6 = -8 + 12 - 10 + 6 = 0$. 5. Since $f(-2) = 0$, $x + 2$ is a factor. --- ### Part 2: Is $2x - 5$ a factor of $16x^4 + 56x^3 + 20x^2 + 8x - 1645$? 6. Rewrite factor as $2x - 5 = 0 ightarrow x = \frac{5}{2}$. 7. Evaluate $f\left(\frac{5}{2}\right) = 16\left(\frac{5}{2}\right)^4 + 56\left(\frac{5}{2}\right)^3 + 20\left(\frac{5}{2}\right)^2 + 8\left(\frac{5}{2}\right) - 1645$. 8. Calculate powers: $$\left(\frac{5}{2}\right)^2 = \frac{25}{4}, \quad \left(\frac{5}{2}\right)^3 = \frac{125}{8}, \quad \left(\frac{5}{2}\right)^4 = \frac{625}{16}$$ 9. Substitute: $$16 \times \frac{625}{16} + 56 \times \frac{125}{8} + 20 \times \frac{25}{4} + 8 \times \frac{5}{2} - 1645$$ 10. Simplify each term: $$625 + 875 + 125 + 20 - 1645 = 1000$$ 11. Since remainder $= 1000 \neq 0$, $2x - 5$ is not a factor. --- ### Part 3: Is $x - 1$ a factor of $x^5 + 5x^4 - 4x^3 + 5x^2 - 6x - 19$? 12. Here $a = 1$. 13. Evaluate $f(1) = 1 + 5 - 4 + 5 - 6 - 19 = -18$. 14. Since remainder $= -18 \neq 0$, $x - 1$ is not a factor. --- **Final answers:** - $x + 2$ is a factor (remainder $0$). - $2x - 5$ is not a factor (remainder $1000$). - $x - 1$ is not a factor (remainder $-18$).