Question

I need help with my Honors project!!

-How do polynomial identities apply to complex numbers?
-How does the binomial theorem use Pascal’s triangle to expand binomials raised to positive integer powers?
-How is the fundamental theorem of algebra true for quadratic polynomials?
-Are rational expressions closed under addition, subtraction, multiplication, and division?

Answer 1

I'll give a medal to anyone who can help me answer these questions!!

Answer 2

Do you need a proof for these, or just an explanation?

Answer 3

Only explanations! @blurbendy

Answer 4

@blurbendy please help me

Answer 5

these are pretty in-depth questions, but ill try to simplify the ideas. let's start with the last one

by definition, a rational expression is any polynomial divided by any polynomial except for zero.

For a rational expression to be "closed" under addition, mult, division, and subtraction, the following must apply:

A rational expression plus a rational expression is a rational expression
A rational expression minus a rational expression is a rational expression
A rational expression times a rational expression is a rational expression
A rational expression divided by a rational expression is a rational expression

let us denote a rational expression as:

a(x) / b(x)

rational expressions are pretty much like fractions when it comes to adding/sub/mult/div
so the following rules apply:

ADDITION: a(x) / b(x) + c(x) / d(x) = (a(x) * d(x) + b(x) * c(x)) / (b(x) * d(x))

SUBTRACTION: a(x) / b(x) - c(x) / d(x) = (a(x) * d(x) - b(x) * c(x)) / (b(x) * d(x))

MULTIPLICATION: a(x) / b(x) * c(x) / d(x) = ( a(x) * c(x )) / ( b(x) * d(x) )

DIVISION: ( a(x) / b(x) ) / ( c(x) / d(x) ) = ( a(x) * d(x) ) / ( b(x) * c(x))

In each case we end up with a polynomial divided by a polynomial which satisfies our definition of a rational expression. Thus, rational expressions are closed under addition, subtraction, multiplication, and division.

It might take you a while to soak that it, but it's not so bad.

Answer 6

So, the correct response would be, yes the ARE closed under all four? I think I understand what you're conveying.

Answer 7

@blurbendy ^

Answer 8

yes, rational expressions are closed under those four operations

Answer 9

any chance you can help me on the other three? you've already done more than enough tbh @blurbendy

Answer 10

yeah, but not right now. this isnt due soon is it?

Answer 11

i should have it done by the end of the week but i would like it done by tomorrow. can you recommend anyone else who might be able to help me? @blurbendy

Answer 12

there's lots of good people on here. I would recommend posting your question again, but just one of your points at a time. people are less likely to help if they feel overwhelmed.

if you don't get an adequate response, I'll be back on in another 3 hours. you can message me or something

Answer 13

thank you so much @blurbendy, i'll do just that

Answer 14

cool

Answer 15

@blurbendy do i have to close this question to ask a new one?

Answer 16

Did you get the answer(s) you needed?

Answer 17

no, i didnt. is there a chance you could still help me out? @blurbendy

Answer 18

I think I have time to help with the first one. As you probably know you can write a complex number in the for (a + bi)

now, let's look at a polynomial identity.

for example, if we have something x^3 - 64, we know that equals x^3 - 4^3 and the polynomial identity says that is equal to (x - 4)(x^2 +4x + 16)

because we know that anything in the form (a^3 - b^3) = (a - b)(a^2 + ab + b^2)

however, we know that we can use complex numbers to factor x^2 + 4x + 16 by using the quadratic formula

after applying the quadratic formula, we get that
x = -2 + 2isqrt(3)
and
x = -2 - 2isqrt(3)

Answer 19

what are the letters "isqrt"? @blurbendy

Answer 20

sqrt stands for square root and i stands for imaginary number

Answer 21

oh, i get it! thank you very much for your help :) @blurbendy

Answer 22

welcome. ill try and come back for the others.