a Consider the Descartes' Rule of Signs to state the zeros.
B
b To draw a graph on a calculator, press the Y= button and type the function in one of the rows. Having written the function, you can push GRAPH to draw it.
C
c To identify the zero, press 2ND and TRACE. Then, choose the second option in the list, and press ENTER. Then, use the right and left arrow keys to move along the curve and determine LeftBound? and RightBound?
A
a 3 positive real zeros or 1 positive real zero and 2 imaginary zeros.
Consider the given polynomial.
M(x)=-2.03x^3+50.1x^2-214x+4020We see the degree of the polynomial is 3. According to the Fundamental Theorem of Algebra, the function has three zeros. This is including any repeated zeros.
Step 2
Let's now determine the type of zeros. To do so, we will examine the number of sign changes for M(x).
There are three sign changes for the coefficients of M(x). According to Descartes' Rule of Signs, the function has one or three positive real zeros. Next, let's write and simplify M(- x).
M(- x)=2.03x^3+50.1x^2+214x+4020
As we did for M(x), we will examine the number of sign changes for M(- x).
There is only one sign change for the coefficients of M(- x). Once again, according to Descartes' Rule of Signs, the function has zero negative real zero. Therefore, there are two options for the types of zeros of M(x). Remember, we know that there are four zeros in total.
Option 1
Option 2
Real Zeros
1 positive
Real Zeros
3 positive
0 negative
0 negative
Imaginary Zeros
2
Imaginary Zeros
Number of Zeros
1+ 0+2=3
Number of Zeros
3+ 0+ =3
b To draw a graph on a calculator, we first press the Y= button and type the function in one of the rows. Having written the function, we can push GRAPH to draw it.
If we are using a standard viewing window, we will need to change the settings so that we can see all of the whole graph.
c Because there is only one x-intercept, there is only one real zero. To identify the zero, we press 2ND and TRACE. Then, we choose the second option in the list, zero and press ENTER.
Next, we will use the right and left arrow keys to move along the curve and determine LeftBound?. The left bound must be on the left of the x-intercept. Then, we press ENTER.
Next, we will determine RightBound? proceeding in the same way. The right bound must be on the right of the x-intercept. Then, we again press ENTER.
Now, we can press ENTER to identify the zero of the function.
The zero of the function is about x=23.8. Because x=0 corresponds to the year 2003, x=23.8 corresponds to the year 2027, approximately. The zero in the context of the situation means that the music hall will not earn any money after 2007.
