1 End behavior and zeros / 3 pts
Let \(f(x)=-x(x-3)^2(x+1)\). (a) Give the degree, the leading coefficient, and the end behavior. (b) List the zeros with their multiplicities and say whether the graph crosses or touches the axis at each.
2 Synthetic division and factoring / 4 pts
Let \(P(x)=x^3-2x^2-5x+6\). (a) Show that \(1\) is a zero. (b) Use synthetic division to divide \(P(x)\) by \(x-1\). (c) Factor \(P(x)\) completely and list its zeros.
3 Remainder Theorem / 3 pts
The polynomial \(Q(x)=2x^3-x^2+kx-4\) leaves remainder \(6\) when divided by \(x-2\). (a) Find \(k\). (b) Find the remainder when \(Q(x)\) is divided by \(x+1\).
4 Real and complex zeros / 4 pts
Let \(R(x)=2x^3-11x^2+14x+10\). (a) List the possible rational zeros. (b) Find a rational zero. (c) Factor \(R(x)\) using synthetic division. (d) Find all complex zeros.
5 Asymptotes and intercepts / 3 pts
Let \(g(x)=\dfrac{4x-8}{x^2-x-6}\). (a) Find the vertical asymptotes. (b) Find the horizontal asymptote. (c) Find the \(x\)- and \(y\)-intercepts.
6 Rational inequality / 3 pts
Using the function \(g\) of the previous exercise, solve \(g(x)\ge0\). Give the answer in interval notation.
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