Respuesta :
Answer:
The polynomial will be P(x) = - 5 (x + 2)²(x - 3)
Step-by-step explanation:
The degree of the polynomial P(x) is 3 and it has zeros at x = - 2 with multiplicity 2 and at x = 3 with multiplicity 1.
Therefore, (x + 2)² and (x - 3) are the factors of the equation.
Let the polynomial is
P(x) = a(x + 2)²(x - 3) ........... (1)
Now, the polynomial passes through the point (2,80).
So, from equation (1) we gat,
80 = a(4)²(-1)
⇒ a = - 5
Therefore, the polynomial will be P(x) = - 5 (x + 2)²(x - 3) (Answer)
The required polynomial is [tex]P(x) = - 5 (x + 2)^{2} (x - 3)[/tex]
Any polynomial have number of roots equal to its degree of polynomial.
Since, the degree of the polynomial P(x) is 3. it means that it has 3 roots.
it has zeros at x = - 2 with multiplicity 2, it means that factor (x - 2) have power 2 and at x = 3 with multiplicity 1 means that factor (x - 3) have power of 1 .
Thus, [tex](x + 2)^{2}[/tex] and (x - 3) are the factors of the equation.
Let us consider the polynomial is [tex]P(x) = k(x + 2)^{2} (x - 3) .[/tex]
Since, the polynomial passes through the point (2,80).
So, substituting point (2, 80) in above polynomial equation.
We get, [tex]80 = a(4)^{2} (-1)[/tex]
a = - 5
Therefore, the polynomial is [tex]P(x) = -5(x + 2)^{2} (x - 3) .[/tex]
Learn more:
https://brainly.com/question/13769924
