You get this to be equal to zero is if at least one of these three Of three different things being equal to zero, the way X equal to zero, so zero is equal to negative four times x plus two times x minus 18. To figure out the x when our height is equal to zero. Seconds after launch, how many seconds that's x, so we want The rocket to hit the ground? That means that the height isĮqual to zero, so if you want to figure out how many Launch will the rocket hit the ground? Pause this video again and This is going to be positive 72 times two, which is 144, so 144 meters. To be negative eight times negative 18, negativeĮight times negative 18, which is the same thing as negative eight times negative nine times two. Which is just going to be two times zero minus 18, To negative four, negative four times zero plus two, Just have to go back to this expression and replaceĪll the xs with zeros. "Well, what is h of zero?" To figure out h of zero, we X is equal to zero, they're essentially saying, Seconds after the launch, so at the time of launch, x Well, what is x at the time of launch? Well, x is the number of Now, the first thing theyĪsk us is what is the height of the rocket at the time of launch? Pause the video and see if Negative four times x plus two times x minus 18. Its height in meters, x secondsĪfter the launch is modeled by h of x is equal to Told a rocket is launched from a platform. So what you really are attempting to do is shift the whole thing 2 units to the left so that you will begin at (0,0) because you do not like the fact that it was launched from a platform. It not that the midpoint is 10 units from the y intercept, it is 10 units from the endpoint 18 and 18-10 = 8. While the -2 point is not within the domain of the word problem, it is still within the graph of the function and is the imaginary reflection of (18.0) which still gives midpoint at 8. The part between 16 and 18 just does not have a part that is reflected across the line of symmetry. Being launched from a platform that is 144 m about the ground gives a starting point at (0,144) not (0,0), and the symmetrical point would be at (16,144) which again gives the same midpoint at 8. So the word problem would cut off all of the parabola less than 0 (including the part that is reflected from 16-18 seconds at -2-0) as well as the part greater than 18. However, within the context of the word problem, the domain would be from 0 to 18 seconds when it hit the ground. If it were a function to be graphed independent of the word problem, both sides of the parabola would extend to negative infinity. In the problem, it clearly states that it is launched from a platform, so your assumption that it must start at 0 meters and 0 seconds is just incorrect. No such general formulas exist for higher degrees.While I like your thinking process, you make several incorrect assumptions. So in conclusion, there are only general formulae for 1st, 2nd, 3rd, and 4th degree polynomials. It's that we will never find such formulae because they simply don't exist. So it's not that we haven't yet found a formula for a degree 5 or higher polynomial. The Abel-Ruffini Theorem establishes that no general formula exists for polynomials of degree 5 or higher. In fact, the highest degree polynomial that we can find a general formula for is 4 (the quartic). Both of these formulas are significantly more complicated and difficult to derive than the 2nd degree quadratic formula! Here is a picture of the full quartic formula:īe sure to scroll down and to the right to see the full formula! It's huge! In practice, there are other more efficient methods that we can employ to solve cubics and quartics that are simpler than plugging in the coefficients into the general formulae. These are the cubic and quartic formulas. There are general formulas for 3rd degree and 4th degree polynomials as well. Similar to how a second degree polynomial is called a quadratic polynomial. A third degree polynomial is called a cubic polynomial. A trinomial is a polynomial with 3 terms. First note, a "trinomial" is not necessarily a third degree polynomial.
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