algebra 2 question

Question

I'm new to algebra and I have no idea what I'm doing on this question.

 

Given the following polynomial, find
a. The zeros and multiplicity of each
b. Where the graph crosses or touches the axis
c. Number of turning points
d. End behavior
f (x) = - (x-3)2 (x +1)3 (x-7)

Answer 1

Hi,

 

I am not sure if the polynomial is multiplied or added or subtracted. I am considering the whole expression as multiplied with 2 and 3 as the powers to (x-3) and (x+1).

 

a) x = 3, -1, and 7 with multiplicity 2, 3 and 1.

b) At x = 3, -1, and 7.

c) 5

d) the graph falls to the left and to the right

 

For more info, contact me for lessons.

 

Best regards,

Gaurav

Answer 2

Zeros---

x=-1,3,7

x=3 has multiplicity 2

 

The graph crosses the x axis at x=-1 & x=7

 

It touches the x axis at x=3

 

Number of turning points---1

 

End behavior --

 

It approaches - infinite as x tends to - infinite

It approaches - infinite as x tends to + infinite

Answer 3

Dear Diane,

Let me work through a similar problem for you.

 

f(x) = -(x+4)²(x+2)(x+7)

 

To find (a), just find values of x that make each equation zero. It helps to seperate them. So, (x+4)² would look like x+4=0. Subtracting 4 from both sides yields a -4. That would be one zero. The multiplicity is how many times a zero shows up. Because x+4 is raised to a power of 2, it has a degree of 2. Therefore the zero (-4) has a multiplicity of 2.

 

For (b), the zeroes are the x-coordinates. Our zero of -4 would represent (-4,0). Now you just need the y-intercept. This is relatively easy. Substitute in zero for all of the x-coordinates, and you are left with the y-intersect! This is because at the y-intersect, the x-coordinate will always be zero. So -(0+4)²(0+2)(0+7)= -224 which is the y-coordinate (0,-224).

 

As for (c), the number of turning points is equal to the degree of the equation minus 1. So for my equation, with degree 4, there would be 3 turning points.

 

And finally for (d). I find it helpful to look at the parent graph for the end behaviors. In a degree 4, the graph looks like a degree 2; it's a parabola. Because My equation is negative, it will be flipped upside down, or inverted across the x-axis in fancy gibberish. My end behavior would be, as f(x) approaches infinity, negative infinity. And for f(x) as it approaches negative infinity, still negative infinity.

 

Hope this helps!