Answer:
a) [tex]x = -1[/tex] and [tex]x = 2[/tex] are roots of the polynomial [tex]g(x) = a\cdot x^{3}+5\cdot x^{2}+4\cdot x + c[/tex].
b) [tex]x = -1[/tex] and [tex]x = 2[/tex] have a multiplicity of 1.
Step-by-step explanation:
a) What are those two zeroes?
The two values of [tex]x[/tex] listed in the table whose outputs are zero are roots of [tex]g(x)[/tex], defined as [tex]g(x) = a\cdot x^{3}+5\cdot x^{2}+4\cdot x + c[/tex]. In other words, those values of [tex]x[/tex] (-1 and 2) satisfies the condition that [tex]g(x) = 0[/tex].
b) What is the multiplicity of each zero?
From the Fundamental Theorem of Algebra, we know that third-order polynomials have three roots and from we understand that those two known zeroes with a multiplicity of 1, since there is a third root within [tex]0 < x < 1[/tex], where [tex]g(0) > 0[/tex] and [tex]g(1) < 0[/tex].
