|
Question 52504: how do you find the real zeros and multiplicity of f(x)=-x^2(x^2-4)(x-5)? How do you determine whether the graph crosses or touches the x-axis at each x-intercept?
Found 2 solutions by venugopalramana, AnlytcPhil:
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! how do you find the real zeros and multiplicity of f(x)=-x^2(x^2-4)(x-5)? How do you determine whether the graph crosses or touches the x-axis at each x-intercept?
find where f(x)=0
we note that f(x)=0.......if x=0
if x^2-4=0...that is (x+2)(x-2)=0......that is x=-2...or....2
if....x-5=0.....that is x=5
so we say it has 4 distint zeros..viz..0,2,-2,5( or roots) and one repeat zero or root equal to '0'of multiplicity =2..since x^2=x*x=0...
if y=0 at more than one point and the curve cuts the x axis at those points,we call those values of x as x intercepts.
if y=0,but the curve has x axis as tangent at that point ,then we say it touches the x axis there.
here the x intercepts are 2,-2,5,where the curve cuts the x axis.the curve touches x axis at x=0
Answer by AnlytcPhil(1806)  (Show Source):
You can put this solution on YOUR website!
how do you find the real zeros and multiplicity of f(x)=-x^2(x^2-4)(x-5)?
How do you determine whether the graph crosses or touches the x-axis at
each x-intercept?
The real zeros of a function are the real values of x which do these two
things:
1. Cause the expression -x²(x²-4)(x-5) to equal to zero.
2. Cause the graph of y = -x²(x²-4)(x-5) to be tangent to (just touch) the
x-axis when the value has an even multiplicity and cross the x-axis when it
has an odd multiplicity.
First we write -x²(x²-4)(x-5) in terms of linear factors, i.e. factors
which only have terms in x¹, or just x (to the first power).
-x²(x²-4)(x-5)
Write the x² as (x)(x) and (x²-4) as its factored form (x-2)(x+2)
-(x)(x)(x-2)(x+2)(x-5)
We set each of these equal to 0.
(We can ignore the negative sign since if any one of those factors are 0,
the annexation of a negative sign cannot affect the result of 0.)
We have 0 as a zero of multiplicity 2, since (x) appears 2 times and 2 is even,
so the graph will TOUCH but not CROSS THROUGH the x-axis where x is 0.
we have 2 as a zero of multiplicity 1, because (x-2) appears only 1 time and 1
is odd, so the graph will CROSS THROUGH the x-axis where x = 2.
we have -2 as a zero of multiplicity 1 , because (x+2) appears only 1 time and
1 is odd, so the graph will CROSS THROUGH the x-axis where x = -2.
we have 5 as a zero of multiplicity 1, because (x-5) appears only one time and
1 is odd, so the graph will CROSS THROUGH the x-axis where x = 5.
Here is the graph:
Edwin
|
|
|
| |
