91Ó°ÊÓ

Determine whether the given algebraic expression is a polynomial. If it is, list its leading coefficient, constant term, and degree. $$1+x^{3}$$

Determine if \(g(x)\) is a factor of \(f(x)\) without using synthetic division or long division. $$f(x)=x^{5}-3 x^{3}-2 x^{2} ; \quad g(x)=x-2$$

Determine if \(g(x)\) is a factor of \(f(x)\) without using synthetic division or long division. $$f(x)=3 x^{3}+5 x^{2}-2 x+3 ; \quad g(x)=x+1$$

Determine if \(g(x)\) is a factor of \(f(x)\) without using synthetic division or long division. $$\begin{aligned} &f(x)=(3+i) x^{3}+(1-2 i) x^{2}+(2+i) x+(1-i)\\\ &g(x)=x-i \end{aligned}$$

Determine whether the given algebraic expression is a polynomial. If it is, list its leading coefficient, constant term, and degree. $$\frac{7}{x^{2}}+\frac{5}{x}-15$$

Find (a) The difference quotient of the function; (b) The vertex of the function's graph; (c) The value of the difference quotient at the \(x\) -coordinate of the vertex. $$g(x)=2 x^{2}-x-1$$

Find the horizontal asymptote, if any, of the graph of the given function. If there is a horizontal asymptote, find a viewing window in which the ends of the graph are within .1 of this asymptote. $$f(x)=\frac{2 x^{3}+4 x^{2}+2 x+1}{3 x^{3}-4 x^{2}-2 x}$$

Analyze the function algebraically. List its vertical asymptotes, holes, y-intercept, and horizontal asymptote, if any. Then sketch a complete graph of the function. $$f(x)=\frac{2 x}{x+1}$$

If the graph of the quadratic function \(h\) is shifted vertically 3 units upward, then reflected in the \(x\) -axis, and then shifted horizontally 5 units to the right, the resulting graph is the parabola \(f(x)=x^{2} .\) What is the rule of the function \(h ?\) What is the vertex of its graph?

Find the rule of the quadratic function whose graph satisfies the given conditions. Vertex at (0,0)\(;\) passes through (2,12)