How Do You Prove Equivalence of Two Polynomials?

[jeremy22511]

Can somebody prove the equivalence statement of two real polynomials in one variable x for me? My Math teacher just told us to remember it as a definition and so I didn't get any proof for it; I attempted to prove it myself and ended up confusing myself with a lot of symbols.

So, can somebody help me with this?

Thanks
Jeremy

 

[micromass]

So, what exactly does this "equivalence statement" state?

 

[HallsofIvy]

The only thing I can think of is that if two polynomials are equal for all x, then they have the same degree and corresponding coeffcients are equal.

If $a_0+ a_1x+ a_2x^2+$$+ \cdot\cdot\cdot\+ a_nx^n=$$b_0+ b_1x+ b_2x^2+ \cdot\cdot\cdot+ b_nx^n$ for all x, then we must have
$(a_0- b_0)+ (a_1- b_1)x$$+ (a_2- b_2)x^2+$$\cdot\cdot\cdot+ (a_n- b_n)x^n= 0$ so it is sufficient to show that if
$a_0+ a_1x+ a_2x^2+ \cdot\cdot\cdot+ a_nx^n= 0$ for all x then $a_0= a_1= a_2= \cdot\cdot\cdot= a_n= 0$. That is, prove that the functions, $1$, $x$, $x^2$, ..., $x^n$ are "independent".

One way to do that is to take n different values for x, say x= 0, 1, 2, ..., n, to get n equations to solve and show that those equations are independent: x= 0 gives $a_0= 0$ so that's easy, x= 1 gives $a_0+ a_1+ a_2+ \cdot\cdot\cdot+ a_n= 0$, x= 2 gives $a_0+ 2a_1+ 4a_2+ \cdot\cdot\cdot 2^n a_n= 0$, etc.

More sophisticated but simpler is to note that if $a_0+ a_1x+ a_2x^2+ \cdot\cdot\cdot+ a_nx^n= 0$ for all x, then it is a constant so its derivative, $a_1+ 2a_2x+ \cdot\cdot\cdot+ na_nx^{n-1}$ i also equal to 0 for all x and so its derivative if 0 for all x, etc. Setting x= 0 in the formula for the polynomial and all of its derivatives gives $a_0= 0$, $a_1= 0$, $2a_2= 0$, ..., $n! a_n= 0$ which again say that all coefficients are 0.

 

[jeremy22511]

Thanks. That really helped.

Jeremy

 

[CellCoree]

Sure, Jeremy. I can provide some insight and explanation on the concept of equivalence of polynomials.

First of all, let's define what we mean by equivalence of polynomials. Two polynomials are considered equivalent if they have the same degree and the same coefficients. In other words, they have the same terms and the same powers of x.

For example, consider the polynomials 3x^2 + 5x + 2 and 2x^2 + 5x + 6. These two polynomials are equivalent because they both have the same degree (2) and the same coefficients (3, 5, and 2 for the first polynomial and 2, 5, and 6 for the second polynomial).

Now, to prove the equivalence statement, we can use the definition we just discussed. We can compare the terms and coefficients of the two polynomials and show that they are the same.

For instance, let's compare the first term of both polynomials, which is 3x^2 and 2x^2. We can see that they both have the same power of x (2) and the coefficients are different (3 and 2). However, we can factor out a common factor of x^2 from both terms, leaving us with 3 and 2, which are indeed the same coefficients.

We can continue this process for all the terms in the polynomials and show that they are equivalent. This is just one approach to proving the equivalence statement, and there are other methods that can be used as well.

I hope this helps clarify the concept for you, Jeremy. It's important to understand the definition of equivalence and how to compare terms and coefficients in order to prove it. Keep practicing and seeking help when needed, and you will become more comfortable with this concept. Good luck!