91Ó°ÊÓ

Perform the division by assuming that \(n\) is a positive integer. $$\frac{x^{3 n}+9 x^{2 n}+27 x^{n}+27}{x^{n}+3}$$

Use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. $$h(x)=\frac{1}{5}(x+2)^{2}(3 x-5)^{2}$$

Cube each complex number. (a) \(-1+\sqrt{3} i\) (b) \(-1-\sqrt{3} i\)

Find the values of \(\boldsymbol{b}\) such that the function has the given maximum or minimum value. $$f(x)=x^{2}+b x-25 ; \text { Minimum value: }-50$$

Perform the division by assuming that \(n\) is a positive integer. $$\frac{x^{3 n}-3 x^{2 n}+5 x^{n}-6}{x^{n}-2}$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=4 x^{3}-3 x^{2}+2 x-1$$

(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. $$f(x)=x^{3}-3 x^{2}+3$$

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

Writing Briefly explain what it means for a divisor to divide evenly into a dividend.

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