Iamhaving a really hard time trying to understand this question. Can someone please show me how to work this question.
Give the following polynomial, find
a. the zeros and the multiplicity of each
b. where the graph crosses or touches the x-axis
c. number of turning points
d. the end behavior
f(x)=(x-1)(x-5)
please show work.
u=x-1 dx=du
v=x-5 dx=dv
f=uv
f'=uv'+vu'=(x-1)+(x-5)=2x-6
zeroes: f(x)=0 at x=1,and x=5
f"=2 means it is always increasing, no inflection point.
end behaviour: at x=+- inf, f(x)=+ undef
one small item:
f" > 0 means its always concave up, not increasing. Its slope is always increasing.
To work through this question, we need to understand a few key concepts related to polynomials.
First, let's define a polynomial: A polynomial is an expression consisting of variables (in this case, x) and coefficients (in this case, 1 and 5), combined using only addition, subtraction, and multiplication, with non-negative integer exponents on the variables.
Now, let's break down the question step by step:
a. Finding the zeros and the multiplicity of each:
The zeros of a polynomial are the values of x where the polynomial is equal to zero. To find the zeros, we set the polynomial equal to zero and solve for x.
In this case, f(x) = (x-1)(x-5) = 0
To find the zeros, we set each factor equal to zero and solve for x:
(x - 1) = 0 => x = 1
(x - 5) = 0 => x = 5
So, the polynomial has two zeros: x = 1 and x = 5.
The multiplicity of a zero represents the number of times a factor is repeated. Since both factors (x - 1) and (x - 5) are distinct linear factors, each with a multiplicity of 1.
b. Finding where the graph crosses or touches the x-axis:
The graph of a polynomial crosses or touches the x-axis at the zeros of the polynomial. In this case, the graph of f(x) crosses the x-axis at x = 1 and x = 5.
c. Finding the number of turning points:
The number of turning points refers to the number of times the graph changes its direction. For a polynomial, the number of turning points is equal to the degree of the polynomial minus one. In this case, the degree of the polynomial is 2 (highest power). Therefore, the number of turning points is 2 - 1 = 1.
d. Determining the end behavior:
The end behavior of a polynomial refers to the behavior of the graph as x approaches positive infinity or negative infinity. To determine the end behavior, we look at the highest power of x in the polynomial and the leading coefficient.
In this case, the highest power is x^2 (degree 2), and the leading coefficient is 1 (coefficient of x^2). When the leading coefficient is positive (in this case, 1), the end behavior is as follows:
- As x approaches positive infinity, the graph of the polynomial goes up (increases without bound).
- As x approaches negative infinity, the graph of the polynomial goes down (decreases without bound).
So, in summary:
a. The zeros are x = 1 and x = 5, both with a multiplicity of 1.
b. The graph of f(x) crosses the x-axis at x = 1 and x = 5.
c. The polynomial has 1 turning point.
d. The end behavior is up and down, as x approaches positive and negative infinity, respectively.