Following a polynomial, find
a. the zero's and the multiplicity of each
b. where the graph crosses or touches the x-axis
c. the number of turning points
d. the end behavior
f(x)=(x+3)(x-5)(x+5)
the zeros are shown above, each of multiplicity 1
the zeros are where the graph crosses the axis, since there are no multiple roots
since the graph crosses the axis 3 times, there are 2 turning points
graph rises up from lower left, turns twice, and exits to upper right
To find the zeros and their multiplicities of the given polynomial f(x) = (x+3)(x-5)(x+5), we can set each factor equal to zero and solve for x.
a. Zeros and their multiplicities:
Setting (x+3) = 0, we get:
x + 3 = 0
x = -3
So, the zero -3 has a multiplicity of 1.
Setting (x-5) = 0, we get:
x - 5 = 0
x = 5
So, the zero 5 has a multiplicity of 1.
Setting (x+5) = 0, we get:
x + 5 = 0
x = -5
So, the zero -5 has a multiplicity of 1.
b. To determine where the graph of f(x) crosses or touches the x-axis, we look at the zeros we found in part a.
The graph crosses the x-axis at x = -3, x = 5, and x = -5.
c. The number of turning points is equal to the number of changes in direction of the graph. For a cubic polynomial like f(x), there can be at most two turning points.
In this case, since f(x) = (x+3)(x-5)(x+5) has three distinct linear factors, the graph of f(x) will have no turning points.
d. To determine the end behavior of the graph, we can look at the highest power term in the polynomial, which is x^3.
As x approaches positive or negative infinity, the sign of x^3 will match the sign of the coefficient in front of x^3, which is positive.
Therefore, the end behavior of the graph of f(x) is:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) approaches negative infinity.
To find the answers to these questions, we first need to understand the factors of the given polynomial. The polynomial f(x) = (x + 3)(x - 5)(x + 5) can be written in factored form as a product of three linear factors.
a. Zeros and their multiplicities:
To find the zeros of the polynomial, we set each factor equal to zero individually and solve for x.
Setting (x + 3) = 0:
x + 3 = 0
x = -3
Setting (x - 5) = 0:
x - 5 = 0
x = 5
Setting (x + 5) = 0:
x + 5 = 0
x = -5
So, the zeros are x = -3, x = 5, and x = -5.
Since each factor is of degree 1, the multiplicity of each zero is 1.
b. Graph crossing or touching the x-axis:
The zeros of the polynomial represent the x-intercepts or the points where the graph of the polynomial crosses or touches the x-axis. In this case, the graph of the polynomial f(x) = (x + 3)(x - 5)(x + 5) will cross the x-axis at the points (-3, 0), (5, 0), and (-5, 0).
c. Number of turning points:
The number of turning points in a polynomial depends on its degree. In this case, the degree of the polynomial is 3, so there will be a maximum of 2 turning points.
d. End behavior:
To determine the end behavior of the polynomial, we look at the leading term, which is the term with the highest power of x. In this case, the leading term is x^3 since the polynomial is in factored form.
If the coefficient of the leading term (in this case, 1) is positive, then the end behavior of the polynomial is as follows:
- As x approaches negative infinity, the polynomial approaches negative infinity.
- As x approaches positive infinity, the polynomial approaches positive infinity.
So, the end behavior of the given polynomial f(x) = (x + 3)(x - 5)(x + 5) is that as x goes to negative infinity, f(x) approaches negative infinity, and as x goes to positive infinity, f(x) approaches positive infinity.