91Ó°ÊÓ

Determine whether the binomial is a factor of the polynomial function. $$ f(x)=2 x^3+5 x^2-37 x-60 ; x-4 $$

Determine whether the binomial is a factor of the polynomial function. $$ h(x)=6 x^5-15 x^4-9 x^3 ; x+3 $$

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. \(-2,3,6\)

Determine whether the binomial is a factor of the polynomial function. $$ g(x)=8 x^5-58 x^4+60 x^3+140 ; x-6 $$

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. \(-4,-2,5\)

ABSTRACT REASONING Suppose \(f(x) \rightarrow \infty\) as \(x \rightarrow-\infty\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow \infty\). Describe the end behavior of \(g(x)=-f(x)\). Justify your answer.

MODELING WITH MATHEMATICS Over a period of 14 years, the number \(N\) of inland lakes infested with zebra mussels in a certain state can be modeled by $$ N=-0.0284 t^4+0.5937 t^3-2.464 t^2+8.33 t-2.5 $$ where \(t\) is time (in years). In which year did the number of infested inland lakes first reach 120 ?

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros. \(-6,0,3-\sqrt{5}\)

In Exercises 43–48, use Pascal’s Triangle to expand the binomial. \((g+2)^5\)

MAKING AN ARGUMENT Your friend claims that \(2-i\) is a complex zero of the polynomial function \(f(x)=x^3-2 x^2+2 x+5 i\), but that its conjugate is not a zero. You claim that both \(2-i\) and its conjugate must be zeros by the Complex Conjugates Theorem. Who is correct? Justify your answer.