Hey just a very beginner question, i've been reading into a book of calculus and this shows up
"a polynomial in $x$ is a sum of constant multiples of
powers of $x$ in which each exponent is zero or a positive integer"
$$P(x) = a_nx^n + a_{n-1}x^{n-1} + … + a_1x + a_0$$
Polynomial is not something new for me, but this definition is, the form i know is:
$$\text{linear} ~:~ ax+b = 0$$
$$\text{quadratic} ~:~ ax^2 + bx + c = 0$$
$$\text{cubic} ~:~ ax^3 + bx^2 + cx + d = 0$$
the question is easy even though I can't relate,
how the above definition can relate to Polynomials ? in what way and why ?
Best Answer
First, polynomials are expressions, not equations. “$ax^2+bx+c = 0$” is an equation. $ax^2+bx+c$ is an expression.
Other than that, those are indeed three kinds of polynomials. The definition needs to be general enough to include those, but also the general form they suggest.
Unpack the given definition one word at a time:
A power of $x$ is an expression like $x^r$. In such an expression, the number $r$ is called the exponent.
A constant is a fixed number. It could be a literal number like $3$ or $\sqrt{2}$, or a symbol for a fixed but unknown number. Usually letters like $a$, $b$, $c$, etc., are used as symbols for constants.
A positive integer is one of the whole numbers $1$, $2$, $3$, ...
I assume you know what sum and multiple are.
Consider the expression $ax^2+bx+c$. It is a sum of three terms: $ax^2$, $bx$, and $c$.
The first is a constant ($a$) multiple of the power $x^2$. The exponent is a positive integer, in this case $2$.
The second is a constant ($b$) multiple of $x$. Since $x=x^1$, it counts as a power with positive integer exponent.
The third is just a constant $c$. But since $x^0 = 1$, we can write $c=cx^0$. We see that $c$ is also equal to a constant multiple of a power of $x$, with exponent $0$.
But now you see how this can generalize. Take any positive whole integer power of $x$, and also the constant $1$. Take any constant multiples of these. Then add them up.
I think polynomials are very tricky to define, even though we can quickly get a picture of what is meant with a few examples. One explicit definition is that a polynomial is an expression of the form $$ a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0 $$ where $n$ is a nonnegative integer, $a_0, \dots, a_n$ are constants, and $a_n \neq 0$. See? “A sum of constant multiples of powers of $x$, where the exponents are positive integers or zero.”
Another, more implicit version is that polynomials are defined by these relations:
I'll leave it as an exercise to you to show that two definitions are equivalent.