Exponents as Coefficients

May 2016
3
0
Americas
Can an exponent ever be considered the coefficient of a variable? For instance, in the expression x^2, what is the coefficient of x? Is it 2, or is it x, since x^2 = x*x?
 

romsek

MHF Helper
Nov 2013
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you wouldn't say that a variable $x$ acts as a coefficient of that same variable.

so in the expression $x\cdot x$ we would not call $x$ the coefficient of the $x$ term.

we would say that in the expression $x^2= x^2 + 0\cdot x$,

$0$ is the coefficient of the $x$ term
 
May 2016
3
0
Americas
Where is this explained. It wouldn't make sense to me that 0 would be the coefficient. That would mean x^2 = x^2*0, which would be 0.
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
Where is this explained. It wouldn't make sense to me that 0 would be the coefficient. That would mean x^2 = x^2*0, which would be 0.
no, the coefficient of the $x^2$ term is 1.

the coefficient of the $x$ term is 0

a sum of the weighted powers of $x$ is called a polynomial in $x$, and the weights are called coefficients

so in general

$p(x) = c_0 + c_1 x + c_2 x^2 + \dots + c_n x^n$

if $p(x)=x^2$ it should be pretty clear that

$c_0=0, c_1=0, c_2=1, c_3=0 \dots c_k=0, k>2$
 
May 2016
3
0
Americas
So in X^2, you say 0 is the coefficient of X, and 1 is the coefficient of the entire expression. How is this? In x = 2, it would be incorrect to say that 0 is the coefficient of X because this would mean X would always mean 0.
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
Polynomials

learn about polynomials, you're just confusing yourself.