Given one factor, find the solutions to the polynomial. Use synthetic or long division. Show work. 3) y=4x^3-12x^2-x+15 factor: 2x-3 y=x^4-8x^3-23x^2+30x factor: x-1

1. Perform Synthetic Division for First Polynomial<br /> Divide $4x^3 - 12x^2 - x + 15$ by $2x - 3$. Use synthetic division with root $\frac{3}{2}$.<br />- Coefficients: $[4, -12, -1, 15]$<br />- Root: $\frac{3}{2}$<br />- Process:<br /> - Bring down the 4.<br /> - Multiply 4 by $\frac{3}{2}$, add to -12: $4 \times \frac{3}{2} = 6$, $-12 + 6 = -6$.<br /> - Multiply -6 by $\frac{3}{2}$, add to -1: $-6 \times \frac{3}{2} = -9$, $-1 + (-9) = -10$.<br /> - Multiply -10 by $\frac{3}{2}$, add to 15: $-10 \times \frac{3}{2} = -15$, $15 + (-15) = 0$.<br /><br />2. Write Quotient and Remainder for First Polynomial<br /> The quotient is $4x^2 - 6x - 10$ with remainder 0. Solve $4x^2 - 6x - 10 = 0$.<br />- Simplify: $2x^2 - 3x - 5 = 0$.<br />- Use quadratic formula: **$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$**.<br />- Here, $a = 2$, $b = -3$, $c = -5$.<br />- Calculate discriminant: $(-3)^2 - 4(2)(-5) = 9 + 40 = 49$.<br />- Solutions: $x = \frac{3 \pm 7}{4}$.<br />- Roots: $x = 2.5$, $x = -1$.<br /><br />3. Perform Synthetic Division for Second Polynomial<br /> Divide $x^4 - 8x^3 - 23x^2 + 30x$ by $x - 1$. Use synthetic division with root $1$.<br />- Coefficients: $[1, -8, -23, 30, 0]$<br />- Root: $1$<br />- Process:<br /> - Bring down the 1.<br /> - Multiply 1 by 1, add to -8: $1 \times 1 = 1$, $-8 + 1 = -7$.<br /> - Multiply -7 by 1, add to -23: $-7 \times 1 = -7$, $-23 + (-7) = -30$.<br /> - Multiply -30 by 1, add to 30: $-30 \times 1 = -30$, $30 + (-30) = 0$.<br /> - Multiply 0 by 1, add to 0: $0 \times 1 = 0$, $0 + 0 = 0$.<br /><br />4. Write Quotient and Remainder for Second Polynomial<br /> The quotient is $x^3 - 7x^2 - 30x$ with remainder 0. Factor further if possible.<br />- Factor out $x$: $x(x^2 - 7x - 30)$.<br />- Solve $x^2 - 7x - 30 = 0$ using quadratic formula.<br />- Here, $a = 1$, $b = -7$, $c = -30$.<br />- Calculate discriminant: $(-7)^2 - 4(1)(-30) = 49 + 120 = 169$.<br />- Solutions: $x = \frac{7 \pm 13}{2}$.<br />- Roots: $x = 10$, $x = -3$.