Solve the following polynomials using synthetic division and then solving the quadratic expression. Show your work. 7) y=x^4+x^3-3x^2+3x-18 8) y=x^4-5x^3-2x^2+19x+15

1. Perform Synthetic Division for Polynomial 7<br /> Use synthetic division to divide $x^4 + x^3 - 3x^2 + 3x - 18$ by a root guess. Assume $x = 2$ is a root.<br /> Coefficients: 1, 1, -3, 3, -18. <br /> Bring down the leading coefficient (1). Multiply by 2 and add to next coefficient repeatedly:<br /> 1 | 1 1 -3 3 -18<br /> | 2 6 6 18<br /> ---------------------<br /> 1 3 3 9 0<br /> Resulting polynomial: $x^3 + 3x^2 + 3x + 9$.<br /><br />2. Solve the Quadratic Expression from Step1<br /> Factor $x^3 + 3x^2 + 3x + 9$. Try $x = -3$ as a root.<br /> Coefficients: 1, 3, 3, 9.<br /> 1 | 1 3 3 9<br /> | -3 0 -9<br /> -----------------<br /> 1 0 3 0<br /> Resulting polynomial: $x^2 + 3$.<br /> Solve $x^2 + 3 = 0$: $x = \pm i\sqrt{3}$.<br /><br />3. Combine Results for Polynomial 7<br /> Roots are $x = 2$, $x = -3$, $x = i\sqrt{3}$, $x = -i\sqrt{3}$.<br /><br />4. Perform Synthetic Division for Polynomial 8<br /> Use synthetic division to divide $x^4 - 5x^3 - 2x^2 + 19x + 15$ by a root guess. Assume $x = 3$ is a root.<br /> Coefficients: 1, -5, -2, 19, 15.<br /> 1 | 1 -5 -2 19 15<br /> | 3 -6 -24 -15<br /> -----------------------<br /> 1 -2 -8 -5 0<br /> Resulting polynomial: $x^3 - 2x^2 - 8x - 5$.<br /><br />5. Solve the Quadratic Expression from Step4<br /> Factor $x^3 - 2x^2 - 8x - 5$. Try $x = 5$ as a root.<br /> Coefficients: 1, -2, -8, -5.<br /> 1 | 1 -2 -8 -5<br /> | 5 15 35<br /> -----------------<br /> 1 3 7 0<br /> Resulting polynomial: $x^2 + 3x + 7$.<br /> Solve $x^2 + 3x + 7 = 0$: $x = \frac{-3 \pm i\sqrt{19}}{2}$.<br /><br />6. Combine Results for Polynomial 8<br /> Roots are $x = 3$, $x = 5$, $x = \frac{-3 + i\sqrt{19}}{2}$, $x = \frac{-3 - i\sqrt{19}}{2}$.