Polynomial & Rational Functions — Quiz 2

Questions in this quiz (6)

1. Three of the roots of a polynomial equation with rational coefficients are $$2$$, $$3i$$, and $$1+\sqrt{5}$$. What is the lowest possible degree of the equation?

   • $$3$$
   • $$4$$
   • $$5$$
   • $$6$$

2. Let $$f(x)$$ represent a third-degree polynomial function with real coefficients. When $$y=f(x)$$ is graphed, it is found that $$f(1)=3$$ and $$f(4)=-2$$. Which statement must be true?

   • The function $$f(x)$$ has exactly one real zero in the interval $$[1,4]$$.
   • The function $$f(x)$$ has at least one real zero in the interval $$[1,4]$$.
   • The function $$f(x)$$ has exactly two real zeros in the interval $$[1,4]$$.
   • The function $$f(x)$$ has no real zeros in the interval $$[1,4]$$.

3. Let $$p(x)$$ represent a fourth-degree polynomial function with real coefficients. When $$y=p(x)$$ is graphed, it is found that $$p(-2)=5$$ and $$p(6)=7$$. Which statement must be true?

   • The function $$p(x)$$ has an odd number of real zeros in the interval $$[-2,6]$$.
   • The function $$p(x)$$ has either no real zeros or an even number of real zeros in the interval $$[-2,6]$$.
   • The function $$p(x)$$ has at least one real zero in the interval $$[-2,6]$$.
   • No conclusion is possible.

4. Given the nonconstant polynomial function $$f(x)$$ whose domain is $$[a,e]$$ where $$a

   • There are three relative extrema in the interval $$[a,e]$$.
   • There are two relative extrema in the interval $$[a,e]$$.
   • $$f(a)<0$$ and $$f(e)<0$$.
   • The graph rises on the left and right.

5. Let $$f$$ be a polynomial function over the interval $$[a,f]$$ with the properties listed in the table below. The average rate of change is decreasing on $$(a,b)$$, decreasing on $$(b,c)$$, increasing on $$(c,d)$$, decreasing on $$(d,e)$$, and increasing on $$(e,f)$$. Which of the following statements must be true?

   • There are two relative extrema.
   • There are three relative extrema.
   • There are two points of inflection.
   • There are three points of inflection.

6. For what value of $$k$$ is $$(x-2)$$ a factor of the function $$f(x)=x^4-5x^3+kx^2+8x-16$$?

   • $$-2$$
   • $$2$$
   • $$4$$
   • $$6$$