Polynomial fundamentals: leading terms, degree, constants, and what makes something a polynomial

A set of QA flashcards covering leading terms, degree, constants, zero polynomials, and what makes something a polynomial, based on the lecture notes.

What is the leading term in a polynomial, and what does the leading coefficient represent, and how is the degree defined?

Leading term: the term with the highest exponent; leading coefficient: the coefficient of that term; degree: the exponent of the leading term.

Why might you include a zero coefficient term like 0x^3 in a polynomial written in descending powers?

To maintain a complete descending sequence of powers of x and to simplify long division or standard form, even if the term's coefficient is zero.

What is the degree of the polynomial 4x^4 + 0x^3 + 5x^2?

4 (the exponent of the leading term x^4).

In g(x) = 3 - (1/2)x, what is the leading term and the degree when arranged in descending powers?

Leading term is -(1/2)x; degree is 1.

For the polynomial 4x^4 + 0x^3 + 5x^2, is there a constant term (an x^0 term)?

No; there is no constant term (the x^0 term is zero or absent).

What are the leading term and degree of h(x) = 9?

Leading term: 9 (the x^0 term); degree: 0.

What is the degree of the zero function f(x) = 0?

There is no degree; the degree is undefined.

Is f(x) = x^(3/2) a polynomial?

No; fractional exponent means it is not a polynomial.

Is f(x) = 1 - 4x^(-1) a polynomial?

No; it has a negative exponent (x in the denominator).

What is the domain of a polynomial function?

All real numbers.

What are two key graph properties of polynomials?

They are smooth, continuous curves with no cusps, gaps, or asymptotes.

What exponents are allowed in polynomials?

Nonnegative integers (zero or positive).

What is the degree of a constant nonzero polynomial, such as f(x) = 9?

0.

Differentiate between the degree of a constant nonzero polynomial and the zero function.

Nonzero constant polynomial has degree 0; the zero function has no degree (undefined).

Why do polynomials require exponents to be in descending order with respect to powers of x?

To clearly identify the leading term and degree, and to ensure a standard form, even when some coefficients are zero.