Answer:
[tex]\begin{gathered} p(\sqrt[]{2})=12\sqrt[]{2}-9 \\ p(-\frac{1}{2})=2.5 \end{gathered}[/tex]
Explanations:
Given the polynomial function
[tex]p(x)=8x^3-6x^2-4x+3[/tex]
We are to find the equivalent value when x = √2 and x = -1/2
a) Substitute x = √2 into the polynomial function to have:
[tex]\begin{gathered} p(\sqrt[]{2})=8(\sqrt[]{2})^3-6(\sqrt[]{2})^2-4(\sqrt[]{2})+3 \\ p(\sqrt[]{2})=8\lbrack(\sqrt[]{2})^2\cdot\sqrt[]{2}\rbrack-6(2)-4\sqrt[]{2}+3 \\ p(\sqrt[]{2})=8(2\sqrt[]{2})-12-4\sqrt[]{2}+3 \\ p(\sqrt[]{2})=16\sqrt[]{2}-4\sqrt[]{2}-12+3 \\ p(\sqrt[]{2})=12\sqrt[]{2}-9 \end{gathered}[/tex]
b) Substitute x = -1/2 into the polynomial function to have:
[tex]\begin{gathered} p(-\frac{1}{2})=8(-\frac{1}{2})^3-6(-\frac{1}{2})^2-4(-\frac{1}{2})+3 \\ p(-\frac{1}{2})=\cancel{8}(-\frac{1}{\cancel{8}})-\cancel{6}^3(\frac{1}{\cancel{4}_2})+\frac{4}{2}+3 \\ p(-\frac{1}{2})=(-1-\frac{3}{2})+(2+3) \\ p(-\frac{1}{2})=-\frac{5}{2}+5 \\ p(-\frac{1}{2})=\frac{5}{2} \\ p(-\frac{1}{2})=2.5 \end{gathered}[/tex]
Therefore the value of the polynomial when x = -1/2 is 2.5
