Polynomials:

1. Determine which functions are polynomial functions. (i) For those that are state whether the polynomial is in standard form or factored form. State the leading term.  (ii) For those that are not state the reason why it is not a polynomial.   solution

(a) \(f(x)=\sqrt{5} x^3+ 4x^5 \)

(b) \(g(x)= x^2(2x+1)(x-3)\)

(c) \(s(t)=\dfrac{t^2-5}{t^3} \)

(d) \(y=\sqrt{x}(2x-1) \)

(e) \(f(v)=\dfrac{9v^2-14v}{3} \)

(f) \(h(t)=t^2(3-t)^2(2t+1) \)

2. Answer the following four questions using the grpah of a polynomial function.   solution

(a) Should the degree of the polynomial be even or odd?   (b) Should the leading coefficient be positive or negative?   (c) Should the multiplicity of the zero \(x = −1\) be even or odd?   (d) Should the multiplicity of the zero \(x = 0\) be even or odd?

3. Given a polynomial \(f(x) = 3x^4+9x^3-30x^2\).   solution

(a) Determine the leading term and end behavior of the graph of the function.

(b) Find the \(x\) and \(y\) intercepts of the function.

(c) Determine the zeros of the function and their multiplicity.  Use this information to determine whether the graph crosses or touches the \(x\)-axis at each \(x\)-intercept.

(d) Draw a graph (rough sketch) of the function.