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## Symmetry of polynomial functions

# Even & odd polynomials

CCSS.Math:

## Video transcript

- [Voiceover] So, we have
three functions here. What I want to do together is think about whether each of these functions are even or odd. And just as a little bit of a reminder, for an even function, if you were to input -x into the function, it's the same thing as inputing x. At f(-x) is the same thing as f(x). For an odd function, if you input -x into the function That's going to be the same thing as a negative of inputting
x into the function. And then, if it's neither, well then neither one of these are going to be true. So let's test this. So let's first look at h(x). And like always, pause the video and see if you can work
through this before I do. So, let's just see what happens if I try to evaluate h(-x). So it's going to be -10 times -x to the 11th power plus
-x to the ninth power minus -x to the third
power plus seven times -x. Well, what is that going to be equal to? Well, -x to the 11th power is going to be equal to the negative
of x to the 11th power. Because we have an odd
exponent right over here. Let me just restate what I just said. So negative, let me just, -x to the third, that's the same thing as the
negative of x to the third. We know it's going to
give you a negative value. Similarly, -x to the
seventh is going to be equal to the negative of x to the seventh. Try it out with some number here. If this was -1, -1 to the seventh power is the same thing as negative times one to the seventh power. So, let me write that over here. So this is going to be
-x to the 11th power. This is going to be -x to the ninth power. And here, while I'm writing this one, we're gonna do the exponents first and then put a negative in front of it. This is going to be negative
of x to the third power, and well this is just -x, so if you have - 10 times the negative of x to the 11th, negative times a negative
is going to be a positive. So this is going to be positive 10x to the 11th power, and then
minus x to the ninth power, minus x to the ninth power, and then you subtract a -x to the third, so that's going to be a
positive x to the third, and then this is going
to be minus seven x. And notice, what we have right over here is similar to what we have up here except all the signs are different. So this is the same thing as if I had took a negative and I multiplied it by h(x), so this is negative h(x). If I took each of these
and I multiplied them by a negative one, then I got what I just got here, which is h(-x). So we just saw that h(-x) is equal to negative h(x), and so we know that this is an odd function. And one telltale signature for it is it's made up of a
bunch of odd functions. We have an odd exponent over here. This is an odd, this is going to be an odd function if it was by itself. This is an odd function
if it was by itself. This is an odd function
if it was by itself. And so is that an odd function by itself. So if you add up a bunch of odd functions, you're going to get an odd function. All of these have odd exponents on them, which make them odd functions. So let's think about the
f(x) right over here. So, f(-x), well negative seven, well -x to the sixth power, you multiply a negative six times, it's
going to be a positive. So that's just going to be negative seven x to the sixth, and then plus three. Well negative x to the fourth power, I don't wanna skip a step here, so this is negative x to the sixth power plus three times negative
x to the fourth power minus nine times negative x squared. And then we have plus eight. Well, a negative number to the sixth power is going to
now be a positive number. So this is going to be equal to negative seven times the
same thing as x to the sixth. Plus, well, negative x to the fourth is same thing as x to the fourth, the negatives all cancel out, so you're gonna have three x to the fourth, then minus, negative x squared is the same thing as x squared, negative three squared is the same thing as three squared,
just giving an example. So it's going to be
negative nine x squared, the negatives here cancel out, plus eight. Well notice, this is
the same thing as f(x). So we have f(-x) is equal to f(x). So f(x) is even. And once again, this should not be a surprise, 'cause it's made up of a bunch of even
functions all added together. Each of these are symmetric about the y-axis, so you add 'em all together, you're going to get an even function. It's made up of a bunch of terms that all have even degrees. So it's the sixth degree, fourth degree, second degree; you could view this as a zero'th degree right over there. Now let's think about g(x). G(x) buried in here. And you might just be able to look at it, and say, "Okay, look, this is "an even function there, this is an "even function, but
this is an odd function, "and this is an odd function." Has a third degree term,
and a first degree term. So it's a mixture of
even and odd functions, so this is gonna be neither even nor odd. And you could test that out. We can look at g(-x) is going to be equal to, well -x to the fourth is the same thing as x to the fourth, so it's gonna be three x to the fourth. Negative x to the third is the negative of positive x to the third. So, that's going to be a
positive 10 x to the third. Actually let me just write
this, let me write it all out. So negative x to the fourth minus 10 times negative x, to the third power, plus -x squared minus -x. So this is just x to the fourth, this is the negative of
positive x to the third, this is the same thing as x squared, and this, well, that's just -x, so it's going to be three
x to the fourth power. Negative 10 times the
negative of x to the third is positive 10 x to the third power plus x squared, and then you
subtract a negative plus x. So notice, this is neither g(x), it's definitely not g(x) 'cause we swapped the signs on the two odd-powered terms, we
swapped the signs over here, but it's also not the negative of g(x). We've only swapped signs on a few of them. And that's because we swapped signs on the odd terms, not on the even terms. So this one right over here is neither even nor odd.