Higher-Degree Polynomials

  1. For each of the following polynomials, identify these features: the degree of the polynomial, the leading coefficient, and the end-behavior (sketch a li’l picture of what the graph of the polynomial looks like to the far left and far right).
    \(7x^2-3x+11\)
    \(3+6x^4-42x^3\)
    \(100-\frac{1}{2}x\)
    \(\pi+x^9\)
    \( \left(3x^3-1\right)\left(4x^4-1\right) \)
    \( \left(1-x^2\right)^5 \)
  2. Perform the necessary operations to write these polynomials in “standard form.”
    • \(\left(x^3+2x^2-1\right) + \left(3x^2-7x+6\right)\)
    • \(\left(x^3+2x^2-1\right) - \left(3x^2-7x+6\right)\)
    • \(\left(x+3\right) \! \left(5x-7\right)\)
    • \(\left(x+3\right) \! \left(x^2+2x\right)\)
  3. The following expression is a cubic polynomial in “factored form.” Write it out in the “standard form” \(ax^3+bx^2+cx+d.\) \[ \left(x-3\right) \left(x-5\right) \left(2x-1\right) \]
  4. The following fifth-degree polynomial has a single root. Use technology to approximate value of this root. Try to be as accurate as your technology will allow. \[ x^{5}-4x^{4}+2x^{3}+4x-5 \]
  5. The following sixth-degree polynomial has two local maximums (humps) and has one local minimum (dip). Use technology to approximate the coordinates of the points where these local minimums and maximums occur. Try to be as accurate as your technology will allow. \[ -x^{6}+x^{5}-7x^{3}+4x^{2}+6x+3 \]
  6. According to data provided by the US Energy Information Administration (EIA) the total US energy supply coming from crude oil products, in quadrillion BTUs (British thermal units), can be modeled by the function \[ B(t) = 0.00001t^4-0.00096t^3+0.0169t^2+0.306t+11.69 \] where \(t\) is the number of years since 1950.
    1. Use technology to visualize the function’s graph \(y = B(t)\) on the domain \(0 \leq t \leq 80.\)
    2. It appears from the graph that energy from crude oil use hit a maximum between 1960 and 1990. According to the model, approximately what year did the maximum occur?
    3. Similarly it appears that crude oil use hit a minimum between 1990 and 2020. According to the model, approximately what year did the minimum occur?
    4. Extrapolating from the data, what does the model indicate our crude oil use in the US to have been last year? Do some research, see if you can find the actual figure for energy from crude oil use in the US last year, and compare it to our model’s prediction
    5. Do you think this model will be an accurate predictor of long-term future crude oil use in the US? Explain why, or why not.
  7. Recall that a polynomial of degree \(n\) may have at most \(n\) roots (\(x\)-intercepts). Can you come up with an example of a cubic (degree three) polynomial with three roots? What about a cubic polynomial with just one root? What about with two roots? What about with no roots at all?

Puzzle

Three friends Anita, Becca, and Charleston are challenged to a game by the Game Maestra. The Game Maestra paints two colored dots on each of their foreheads and tells them that each dot is either blue or yellow, but neither color is used more than four times. She then places the three friends in a circle so that each of them can see the dots on their friends’ foreheads, but not the dots on their own. The game proceeds like this: The Maestra will ask the friends in turn, first Anita, then Becca, then Charleston, then Anita again, then Becca again, and so on, if they know the colors of the dots on their foreheads. When someone responds “no,” the Maestra asks the next person. If someone responds “yes” and is right, the friends win! However if someone responds “yes” and is wrong, all three friends will be banished to the shadow realm for eternity.

The friends were given no time to strategize, but they begin playing. Their responses in turn are “no,” “no,” “no,” “no,” “yes,” and the three friends win! Whare are the colors of the dots on Becca’s forehead?