Sec 2.6: 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. Describe the end behavior of each of the following polynomial functions.   solution

(a) $P(x)=x(x^2-10)$   (b) $P(x)=3x^4-2$   (c) $P(x)=-x^5+5x^3-2x+1$

3. 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

4. Factor the polynomial and use the factored form to find the zeros. Draw a graph (rough sketch) of the function.   solution

(a) $P(x)=x^3-3x^2-10x$   (b) $P(x)= x^4-4x^3+3x^2$   (c) $P(x)=-x^3+2x^2+15x$

5. 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.