Let $f(x)$ be a polynomial such that $f(\cos \theta) = \cos(4\theta)$ for all $\theta$. find $f(x)$. (this is essentially the same as finding $\cos(4\theta)$ in terms of $\cos \theta$; we structure the problem this way so that you can answer as a polynomial. be sure to write your polynomial with the terms in order of decreasing degree.)

Listed in the table is the percentage of students who chose each kind of juice at lunchtime. use the table to determine the measure of the central angle you would draw to represent orange juice in a circle graph

In tossing one coin 10 times, what are your chances for tossing a head? a tail? 2. in tossing one coin 100 times, what are your chances for tossing a head? a tail? 3. in tossing one coin 200 times, what are your chances for tossing a head? a tail? deviation = ((absolute value of the difference between expected heads and observed heads) + (absolute value of the difference between expected tails and observed tails)) divided by total number of tosses. this value should always be positive. 4. what is the deviation for 10 tosses? 5. what is the deviation for the 100 tosses? 6. what is the deviation for 200 tosses? 7. how does increasing the total number of coin tosses from 10 to 100 affect the deviation? 8. how does increasing the total number of tosses from 100 to 200 affect the deviation? 9. what two important probability principles were established in this exercise? 10. the percent of occurrence is the obtained results divided by the total tosses and multiplied by 100%. toss the coins 100 times and record your results. calculate the percent occurrence for each combination. percent head-head occurrence: percent tail-tail occurrence: percent head-tail occurrence:

If t8 = 4 and t12 = −2, find the first three terms of the arithmetic sequence.