Answer:
Equation of the polynomial function is
[tex]f(x)=\frac{1}{6}( x^3-10x^2+31x-30)[/tex]
The y-intercept is -5
Step-by-step explanation:
According to the Descarte's theorem , a polynomial p(x) that has a zero r, that is p(r)=0, then it follows (x-r) is a factor of p.
If f(x) has zeros 2,3 and 5 then f(x) has factors (x-2),(x-3) and (x-5) hence to find the equation
f(x)=a(x-2)(x-3)(x-5)
f(x)={a(x-2)(x-3)(x-5)}-----------------------expand
f(x)={a (x(x-3)-2(x-3)⇒x²-3x-2x+6⇒x²-5x+6
f(x)={a (x²-5x+6)(x-5)}
f(x)={a x(x²-5x+6)-5(x²-5x+6)}
f(x)=a(x³-5x²+6x-5x²+25x-30)
Given that we have one hang point (0,-5), let x=0, f(x)=-5 find a
-5=a(0-0+0-30)
-5=-30a
a=-5/-30
a=1/6
Equation⇒ f(x)=1/6 (x³-10x²+31x-30)
To get y-intercept plugin x=0 in the equation , you notice the value of the function will be -5, hence y-intercept is -5
