91Ó°ÊÓ

Write the quadratic function $$f(x)=a x^{2}+b x+c$$ in standard form to verify that the vertex occurs at $$\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=3 x^{3}+2 x^{2}+x+3$$

Determine whether the statement is true or false. Justify your answer.\(-i \sqrt{6}\) is a solution of \(x^{4}-x^{2}+14=56\).

Using the Intermediate Value Theorem (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. $$f(x)=0.11 x^{3}-2.07 x^{2}+9.81 x-6.88$$

Briefly explain how to check polynomial division, and justify your reasoning. Give an example.

Determine whether the statement is true or false. Justify your answer.$$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$.

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}+3 x^{2}-2 x+1\) (a) Upper: \(x=1\) (b) Lower: \(x=-4\)

Assume that the function $$f(x)=a x^{2}+b x+c, \quad a \neq 0$$ has two real zeros. Prove that the \(x\) -coordinate of the vertex of the graph is the average of the zeros of \(f\) (Hint: Use the Quadratic Formula.)

Using the Intermediate Value Theorem (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. $$h(x)=x^{4}-10 x^{2}+3$$