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Answer:
f(r) = (x -1)(x -4+√7)(x -4-√7)
Step-by-step explanation:
The signs of the terms are + - + -. There are 3 changes in sign, so Descartes' rule of signs tells you there are 3 or 1 positive real roots.
The rational roots, if any, will be factors of 9, the constant term. The sum of coefficients is 1 -9 +17 -9 = 0, so you know that r=1 is one solution to f(r) = 0. That means (r -1) is a factor of the function.
Using polynomial long division, synthetic division (2nd attachment), or other means, you can find the remaining quadratic factor to be r^2 -8r +9. The roots of this can be found by various means, including completing the square:
r^2 -8r +9 = (r^2 -8r +16) +9 -16 = (r -4)^2 -7
This is zero when ...
(r -4)^2 = 7
r -4 = ±√7
r = 4±√7
Now, we know the zeros are {1, 4+√7, 4-√7), so we can write the linear factorization as ...
f(r) = (r -1)(r -4 -√7)(r -4 +√7)
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<em>Comment on the graph</em>
I like to find the roots of higher-degree polynomials using a graphing calculator. The red curve is the cubic. Its only rational root is r=1. By dividing the function by the known factor, we have a quadratic. The graphing calculator shows its vertex, so we know immediately what the vertex form of the quadratic factor is. The linear factors are easily found from that, as we show above. (This is the "other means" we used to find the quadratic roots.)
Answer:
(a) The proportion of tenth graders reading at or below the eighth grade level is 0.1673.
(b) The 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is (0.198, 0.260).
Step-by-step explanation:
Let <em>X</em> = number of students who read above the eighth grade level.
(a)
A sample of <em>n</em> = 269 students are selected. Of these 269 students, <em>X</em> = 224 students who can read above the eighth grade level.
Compute the proportion of students who can read above the eighth grade level as follows:
The proportion of students who can read above the eighth grade level is 0.8327.
Compute the proportion of tenth graders reading at or below the eighth grade level as follows:
Thus, the proportion of tenth graders reading at or below the eighth grade level is 0.1673.
(b)
the information provided is:
<em>n</em> = 709
<em>X</em> = 546
Compute the sample proportion of tenth graders reading at or below the eighth grade level as follows:
The critical value of <em>z</em> for 95% confidence interval is:
Compute the 95% confidence interval for the population proportion as follows:
Thus, the 95% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is (0.198, 0.260).