91Ó°ÊÓ

Determine whether the given value is a zero of the function. $$f(x)=3 x-2 ; x=2 / 3$$

You are given a polynomial equation \(f(x)=0 .\) Specify the multiplicity of each repeated root. Then use a graphing utility to visually verify that the graph of \(y=f(x)\) is tangent to the \(x\) -axis at each repeated root. 17\. (a) \((x+1)^{2}(x+2)=0\) (b) \((x+1)(x+2)^{3}=0\) (c) \((x+1)^{2}(x+2)^{3}=0\)

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of \(x^{2}\) is 1. Use either of the methods shown in Example 3 $$r_{1}=\frac{1}{2}+\frac{1}{4} \sqrt{5}$$

Use Descartes's rule of signs to obtain information regarding the roots of the equations. $$5 x^{4}+2 x-7=0$$

Use the rational roots theorem and the remainder theorem to determine the roots of the equation \(x^{3}+2 x^{2}-5 x-6=0 .\) (This is to verify a statement made in Example \(2 .\) )

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{x^{5}-10 x^{4}+36 x^{3}-55 x^{2}+32 x+1}{x^{4}-6 x^{3}+12 x^{2}-8 x}$$

Use the fact that \(i^{4}=1\) to simplify each expression (as in Example \(5(b)]\). $$i^{36}$$

Simplify each expression. $$\sqrt{-20}-3 \sqrt{-45}+\sqrt{-80}$$