EWA Resampling All the images are exactly the same distortion, just using different 're-sampling' techniques. The last image in the above used the default EWA settings of the Generalized Distortion Operatorand as you can see it produced an extremely high quality result. However it took 4. The first image has the default EWA resampling turned off by using a " -filter point" setting.
Factoring quadratics Video transcript We're asked to factor 4y squared plus 4y, minus And whenever you have an expression like this, where you have a non-one coefficient on the y squared, or on the second degree term-- it could have been an x squared-- the best way to do this is by grouping.
And to factor by grouping we need to look for two numbers whose product is equal to 4 times negative So we're looking for two numbers whose product-- let's call those a and b-- is going to be equal to 4 times negative 15, or negative And the sum of those two numbers, a plus b, needs to be equal to this 4 right there.
So let's think about all the factors of negative 60, or And we're looking for ones that are essentially 4 apart, because the numbers are going to be of different signs, because their product is negative, so when you take two numbers of different signs and you sum them, you kind of view it as the difference of their absolute values.
If that confuses you, don't worry about it. But this tells you that the numbers, since they're going to be of different size, their absolute values are going to be roughly 4 apart. So we could try out things like 5 and 12, 5 and negative 12, because one has to be negative. If you add these two you get negative 7, if you did negative 5 and 12 you'd get positive 7.
They're just still too far apart. What if we tried 6 and negative 10?
Then you get a negative 4, if you added these two. But we want a positive 4, so let's do negative 6 and Negative 6 plus 10 is positive 4.
So those will be our two numbers, negative 6 and positive Now, what we want to do is we want to break up this middle term here. The whole point of figuring out the negative 6 and the 10 is to break up the 4y into a negative 6y and a 10y.
So let's do that.
So this 4y can be rewritten as negative 6y plus 10y, right? Because if you add those you get 4y. And then the other sides of it, you have your 4y squared, your 4y squared and then you have your minus All I did is expand this into these two numbers as being the coefficients on the y. If you add these, you get the 4y again.
Now, this is where the grouping comes in. You group the term. Let me do it in a different color. So if I take these two guys, what can I factor out of those two guys?
Well, there's a common factor, it looks like there's a common factor of 2y.
So if we factor out 2y, we get 2y times 4y squared, divided by 2y is 2y. And then negative 6y divided by 2y is negative 3. So this group gets factored into 2y times 2y, minus 3.
Now, let's look at this other group right here. This was the whole point about breaking it up like this. And in other videos I've explained why this works.
Now here, the greatest common factor is a 5.Leading coefficient definition, the coefficient of the term of highest degree in a given polynomial.
5 is the leading coefficient in 5x3 + 3x2 − 2x + 1. See more. License. The materials (math glossary) on this web site are legally licensed to all schools and students in the following states only: Hawaii. If you're behind a web filter, please make sure that the domains *benjaminpohle.com and *benjaminpohle.com are unblocked.
In this tutorial we will be taking a close look at finding zeros of polynomial functions. We will be using things like the Rational Zero Theorem and Descartes's Rule of Signs to help us through these problems. In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
An example of a polynomial of a single indeterminate, x, is x 2 − 4x + benjaminpohle.com example in three variables is x 3 + 2xyz 2 − yz + 1. License. The materials (math glossary) on this web site are legally licensed to all schools and students in the following states only: Hawaii.