contestada

Find the polynomial f(x) that has the roots of -2, 5 of multiplicity 2. Explain how you would verify the zeros of f(x).

Respuesta :

If -2 is a root of f, then (x-(-2))=(x+2) is a factor of f

similarly, if 5 is a root of f, then (x-5) is a factor of f(x).

These roots have multiplicity 2 means that there are 2 of each,

so f(x) is written as (x+2)(x+2)(x-5)(x-5)

[tex]f(x)= (x+2)^{2} (x-5)^{2} [/tex]

[the most general form is f(x)= c*(x+2)^{2} (x-5)^{2}*P(x), where c is a constant and P(x) another polynomial]

To show that -2 and 5 are zeros of f, we must prove that f(-2)=0 and f(5)=0,

Using the function we wrote:

[tex]f(-2)= (-2+2)^{2} (-2-5)^{2}=0* (-7)^{2}=0[/tex]

similarly:

[tex]f(5)= (5+2)^{2} (5-5)^{2}=0 [/tex]

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