We are asked to complete the following definition.
A polynomial function of degree n and leading coefficient a_n is a function of the form f(x)=a_n x^n+a_(n-1)+... + a_2x^2+ a_1x+a_0, a_n ≠0, where n is a and a_n,a_(n-1),... a_2,a_1,a_0 are numbers.
An example of a polynomial function is a quadratic function. We know that this is a quadratic function because the highest power of the variable x is 2.
Example of a Polynomial [0.5 em]
f(x)=4x^2+2x+7
The highest power of a variable in a polynomial is called its degree. The degree and any other power of a variable in a polynomial must be a nonnegative integer.
Function
Is It a Polynomial?
f(x)=sqrt(3) x^(500)-2/3x^(17)
Yes, polynomial with degree 500. âś“
f(x)=x^(sqrt(3))+x^(Ď€) +x^(1/3)
No, because the powers of the variables are not integers. *
The numbers that the variable terms are multiplied by are called coefficients. A coefficient can be any real number.
Polynomial
Coefficient of x^(81)
Coefficient of x^7
Coefficient of x^1=x
Coefficient of x^0=1
f(x)= π x^(81)-( sqrt(3)/sqrt(2))x^7+ 12x+ - 2
Ď€
sqrt(3)/sqrt(2)
12
-2
Now we are ready to complete the given definition.
A polynomial function of degree n and leading coefficient a_n is a function of the form f(x)=a_n x^n+a_(n-1)+... + a_2x^2+a_1x+a_0, a_n ≠0, where n is a nonnegative integer and a_n,a_(n-1),... a_2,a_1,a_0 are real numbers.
