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# Even & odd polynomials

CCSS.Math:

## Video transcript

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 negative X into the function it's the same thing as inputting X that F of negative x is the same thing as f of X for an odd function if you input negative x into the function that's going to be the same thing as the 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 do let's first look at H of X and like always pause the video and see if you can if you can work through this before I do so let's just see what happens if I try to evaluate H of negative X so it's going to be negative 10 times negative x to the 11th power plus negative X to the ninth power minus negative x to the third power plus seven times negative X well what is that going to be equal to well negative 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 so negative let me just so negative x to the third that's the same thing 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 negative x to the seventh is going to be equal to the negative of X is seven try it out with some number here if this was negative one negative 1 to the seventh power is the same thing as negative times 1 to the seventh power so let me let me write that over here so this is going to be negative X to the 11th power this is going to be negative X to the ninth power and here when I'm writing this way we're going to do the exponents first and then put a negative in front of it this is going to be this is going to be negative of X to the third power and this is just well this is just negative x so if you have negative 10 times the negative of X to the 11th negative times a little negative is going to be a positive so this is going to be positive 10 X to the 11th power and then minus X to the 9th power minus X to the ninth power and then you subtract a negative x to the third so that's going to be a positive x to the third and then this is going to be minus 7x 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 took a negative and I multiplied it by H of X so this is negative H of X so if I took each of these and I multiplied it by a negative one then I got what I just got here which is H of negative x so we just saw that H of negative x is equal to negative H of X and so we know that this is an odd 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 I have odd exponents on them which make them odd functions so let's think about the f of X right over here so F of negative x well negative 7 well negative 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 let me don't want to skip steps here so this is negative X to the sixth power plus three times negative negative x to the fourth power minus 9 times negative x squared and then we have plus eight well negative negative number two the six 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 the same thing as X to the fourth the negatives all cancel out so you're gonna have 3x to the fourth then minus this 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 negative to your cancel out plus eight well notice this is the same thing as f of X so we have F of negative x is equal to f of X so f of X is even and once again this should not be a surprise because 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 them 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 a six degree fourth degree second degree you could view this as a zeroth degree right over there now let's think about G of X G of X buried in here and you might just be able to look at it 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 it has a third degree term and a first degree term so it's a mixture of you know even an odd function so if I so this is going to be neither even nor odd and you could test that out we can look at G of negative x is going to be equal to well negative x to the fourth is the same thing as X to the fourth so it's going to 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 that's going to be a positive 10 X to the third that then 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 negative x squared minus negative 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 then this will that's just negative x so this is going to be 3 X to the fourth power negative 10 times the negative of X to the 3rd is positive 10 X to the 3rd power plus x squared and then you subtract a negative plus X so notice this is neither G of X it's definitely not G of X because we swap the signs on the two odd powered terms we swap the signs over here but it's also not it's also not the negative of G of X we've only swapped signs on a few of them and that's because we've swap signs on the odd ones on the odd terms not on the even terms so this one right over here is neither even nor odd