According to the fundamental theorem of algebra, how many zeros does the polynomial below have?

Question

11. According to the fundamental theorem of algebra, how many zeros does the polynomial below have?
                             F(x)x^4+5x^3+10x^2+20x+24

12. If you know that 7 + 2i is a root of a polynomial function, what is another root of that same function?

13. Give the equation of the horizontal asymptote of the function shown below.
                             G(x)=7x^2/x^2+5
y = 7
x = 7
y = 5
y = 7/5

14. What is the equation for the vertical asymptote of the function shown below?
                              F(x)=5x^4-7/x+3
x = -3
x = 3
x = 5
x = 7/3



15. Which of the following correctly describes the domain of the function shown below?
                                             R(x)=2x/x^2-1
{x:all.real.numbers}
{x:x ≠ 0}
{x:x≠1}
{x:x≠±1}

Answer 1

11. Polynomials always have as many roots as their order. This is a 4th order polynomial- hence 4 roots!

 

12. Complex roots of polynomials always come in pairs. The quadratic equation has that "+/-" in it that will always generate m+ni and m-ni, so 7-2i must be a root of this polynomial equation!

 

13. The order of the numerator and denominator are both 2. so the horizontal asymptote is y=a2/b2 =7.

 

14. Vertical asymptotes are the forbidden x-values. F(x) becomes undefined at x=-3 because this value will cause a zero denominator.

 

15. A graph of R(x) will have vertical asymptotes at ±1 because these values will make the denominator zero. So the domain of the function cannot include ±1, and the answer is {x:x≠±1}.