identify the roots and y - intercept of the function below. fill in the sign table and sketch a graph. your…

identify the roots and y - intercept of the function below. fill in the sign table and sketch a graph. your graph must accurately cross all known intercepts. (f(x)=x(x + 5)^{2}(2x - 6)^{2}(x - 5)) identify the y - intercept of the function. 0 identify all real roots. use commas to separate.

identify the roots and y - intercept of the function below. fill in the sign table and sketch a graph. your graph must accurately cross all known intercepts. (f(x)=x(x + 5)^{2}(2x - 6)^{2}(x - 5)) identify the y - intercept of the function. 0 identify all real roots. use commas to separate.

Answer

Explanation:

Step1: Find the roots

Set $f(x)=0$. Then $x(x + 5)^2(2x - 6)^2(x - 5)=0$. By the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x=0$ or $(x + 5)^2=0$ (which gives $x=-5$) or $(2x - 6)^2=0$ (solving $2x-6 = 0$ gives $x = 3$) or $x - 5=0$ (which gives $x = 5$).

Step2: Determine multiplicity

The root $x = 0$ has multiplicity 1, $x=-5$ has multiplicity 2, $x = 3$ has multiplicity 2 and $x = 5$ has multiplicity 1.

Answer:

$0,-5,3,5$