Covers the fundamental definition of a polynomial function, including its standard form, degree, and leading coefficient. Essential for understanding the building blocks of polynomial analysis.
What is a polynomial function in one variable, \(x\)?
A function that can be written in the form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), where \(n\) is a non-negative integer, and the coefficients \(a_n, a_{n-1}, \dots, a_0\) are real numbers.
What is the 'standard form' of a polynomial?
Writing the terms of the polynomial in descending order of their exponents. For example, \(3x^2 - 5x^4 + 7\) in standard form is \(-5x^4 + 3x^2 + 7\).
How do you determine the 'degree' of a polynomial?
The degree is the highest exponent of the variable in the polynomial after it has been written in standard form. The degree of \(-5x^4 + 3x^2 + 7\) is \(4\).
What is the 'leading coefficient' of a polynomial?
It is the coefficient of the term with the highest exponent when the polynomial is in standard form. For \(-5x^4 + 3x^2 + 7\), the leading coefficient is \(-5\).
How are polynomials classified by their degree? Give examples for degrees \(0\) through \(4\).
How do you evaluate a polynomial function, such as \(P(x) = 2x^3 - x^2 + 5\), for a given value, say \(x = -2\)?
Substitute the value for \(x\) into the function. \(P(-2) = 2(-2)^3 - (-2)^2 + 5 = 2(-8) - 4 + 5 = -16 - 4 + 5 = -15\).
Is \(f(x) = 3x^2 + 4x^{-1} - 5\) a polynomial function?
No. A polynomial function cannot have terms with negative exponents. The term \(4x^{-1}\) violates the definition.
Is \(g(x) = \sqrt{x} + 7\) a polynomial function?
No. A polynomial function cannot have terms with fractional exponents. The term \(\sqrt{x}\) is equivalent to \(x^{1/2}\), which violates the definition that exponents must be non-negative integers.
