What is the completely factored form of d4 -8d2 + 16
1 answer:
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Answer:
The 96% confidence interval for the population proportion of customers satisfied with their new computer is (0.77, 0.83).
Step-by-step explanation:
We have to calculate a 96% confidence interval for the proportion.
We consider the sample size to be the customers that responded the survey (n=800), as we can not assume the answer for the ones that did not answer.
The sample proportion is p=0.8.
The standard error of the proportion is:
The critical z-value for a 96% confidence interval is z=2.054.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
The 96% confidence interval for the population proportion is (0.77, 0.83).
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.)
