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 + 5)(x + 2)^2(x - 1)(x - 3) identify the y - intercept of the function. 60 identify all real roots. use commas to separate. -5,-2,1,3 fill in the sign table. click an area to change its sign.

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 + 5)(x + 2)^2(x - 1)(x - 3) identify the y - intercept of the function. 60 identify all real roots. use commas to separate. -5,-2,1,3 fill in the sign table. click an area to change its sign.

Answer

Explanation:

Step1: Find y - intercept

Set $x = 0$ in $f(x)=(x + 5)(x + 2)^2(x - 1)(x - 3)$. Then $f(0)=(0 + 5)(0+ 2)^2(0 - 1)(0 - 3)=5\times4\times(-1)\times(-3)=60$.

Step2: Find real roots

Set $f(x)=0$. Since $f(x)=(x + 5)(x + 2)^2(x - 1)(x - 3)=0$, then $x+5 = 0$ gives $x=-5$, $(x + 2)^2=0$ gives $x=-2$, $x - 1=0$ gives $x=1$ and $x - 3=0$ gives $x=3$.

Step3: Fill sign - table

  1. Consider the intervals $(-\infty,-5)$, $(-5,-2)$, $(-2,1)$, $(1,3)$ and $(3,\infty)$.
    • For $x<-5$, let $x=-6$. Then $f(-6)=(-6 + 5)(-6 + 2)^2(-6 - 1)(-6 - 3)=(-1)\times16\times(-7)\times(-9)<0$.
    • For $-5<x<-2$, let $x=-3$. Then $f(-3)=(-3 + 5)(-3 + 2)^2(-3 - 1)(-3 - 3)=2\times1\times(-4)\times(-6)>0$.
    • For $-2<x<1$, let $x = 0$. Then $f(0)=60>0$.
    • For $1<x<3$, let $x = 2$. Then $f(2)=(2 + 5)(2 + 2)^2(2 - 1)(2 - 3)=7\times16\times1\times(-1)<0$.
    • For $x>3$, let $x = 4$. Then $f(4)=(4 + 5)(4 + 2)^2(4 - 1)(4 - 3)=9\times36\times3\times1>0$.

The sign - table:

Interval Sign of $f(x)$
$(-\infty,-5)$ -
$(-5,-2)$ +
$(-2,1)$ +
$(1,3)$ -
$(3,\infty)$ +

Answer:

The y - intercept is $60$. The real roots are $x=-5,x=-2,x = 1,x = 3$. The sign - table is as shown above.