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## Algebra (all content)

### Course: Algebra (all content)>Unit 10

Lesson 37: Symmetry of polynomial functions

# Even & odd polynomials

Sal analyzes three different polynomials to see if they are even, odd, or neither.

## Want to join the conversation?

• Doesn't all even functions have all even powers; and all odd functions have all odd powers; and if the function is neither odd or even, then the function have both even and odd functions?
• Yes that is a good observation. All even polynomial functions have only even powers in their complete expansion and all odd polynomial functions only have odd powers in their complete expansion and completely expanded polynomial functions that are neither have both.
• f(x)=x ^5 - x^3 + 7
odd, even, or neither?

The correct answer is neither odd or even. But why? Is this supposed to be odd, since all the powers are odd.
• There's an easily-overlooked fact about constant terms (the 7 in this case).
A constant, C, counts as an even power of x, since C = Cx^0 and zero is an even number.

So in this case you have
x^5: (odd)
x^3: (odd)
7: (even)
So you have a mix of odds and evens, hence the function is neither.
• a polynomial with factors of one and itself is called a/an ?
• Hello. Is there a possibility that there will be a term for a function that is neither an even nor odd?
• So if at least one of the terms is even in an otherwise odd expression, will it be neither odd nor even?
(1 vote)
• If all terms are even expressions, then the function is an even function. If all terms are odd, then the function is an odd function. If Some terms are even and some terms are odd, then the function is neither even nor odd.
• I basically look at wither the exponents are either positive or negative is this correct?
(1 vote)
• Yep. If it's all even it's even, If it's all odd it's odd, If it's a mix it's neither.
• I have a problem that says there's a function h(x) that's both even and odd. There's these other two functions:

The function f(x) is defined by f(x) = ax^2 + bx + c . Another function g(x) is defined as g(x) = psin(x) + qx + r, where a, b, c, p, q, r are real constants.

Given that f(x) is an even function, show that b = 0. (I've done this)

Given that g(x) is an odd function, find the value of r. (I've done this too)

A function h(x) is both odd and even, with a domain of all real numbers. Find h(x).

I tried h(-x) = h(x) = -h(x), but I stopped there.
I tried to solve f(g(x)) and g(f(x)) but realized it was too much for a 2-point question (this is an IB problem). Is there a right answer in this, and does it involve f(x) and/or g(x)?
(1 vote)
• If h is even then h(x) = h(-x), but if h is odd h(-x) = -h(x)
Let h(x) = a
then h(-x) = a (even)
and h(-x) = -a (odd)
Therefore a = -a, and a can only be 0
So h(x) = 0
If you think about this graphically, what is the only line (defined for all reals) that can be both mirror symmetric about the y-axis (even) and rotationally symmetric about the origin (odd).
I don't think f and g are involved.
• I am a student. Last year I was learning math on KA. I used to take note of all that I had learnt. I started learning Polynomials , but then found out that there was so much to learn about it. So i quit Polynomials as I thought I would learn it later. However I did take note of all that I started with Polynomials and left blank pages to fill later. Last year there were tons of videos on Polynomials. But after the interface of KA changed I don't find as much content on Polynomials as there was last year. Even though I completed the entire polynomials page in Algebra All Content, I don't find anything new to take note of. So all the blank pages that I left last year remains. Has KA deleted all the content of Polynomials as they switched to new Interface?
(1 vote)
• If you were looking at "Algebra All Content", most of the content is now in either Algebra 1 or Algebra 2. I have found that some of the content that was only ever at the "all content" level can still be found if you search for it by topic or name.
Hope this helps.
• If I added a number to the h(x) function, it would not be an odd function any more, right? For example if I add 2, then the function shifts upwards and is not symmetric with respect to the origin.
(1 vote)
• Yes, this is exactly right. Translating an already odd function will cause it not to be symmetrical anymore. This isn't the case with even functions, however, because whether you move the function up or down doesn't affect its symmetry.