Polynomials and Polynomial Functions — Integrated Mathematics

10.1 Polynomials and Polynomial Functions

This page sets out what a polynomial is, the names for its parts, and some first computations with coefficients.

Definition 10.1.1 (Polynomials and polynomial functions). is a polynomial function , and the right-hand side is the polynomial. Each summand () is called a term. When the highest power of is , the polynomial is a constant (e.g. ); the zero polynomial is the constant . All others are nonzero. is the degree of the polynomial function : the highest power of in it.

Written with the exponents descending from left to right, has

Leading term: .
Constant term: .
Domain: all real numbers.

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic

Number of terms	Name
1	monomial
2	binomial
3	trinomial

Note. The exponents are all non-negative integers. Order by descending exponents when regarding polynomials.

In each example below, decide whether the function is a polynomial and, where it is, find its degree, leading term and constant term.

Example 10.1.2. Find the degree, leading term and constant term of .

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Reorder: . Remember, this is . So the degree is , the leading term is , and the constant term is .

Example 10.1.3. Find the degree, leading term and constant term of .

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Reorder: . The degree is , the leading term is , and the constant term is .

Example 10.1.4. .

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Degree : this is a constant function, which is a polynomial of degree .

Example 10.1.5. .

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No degree.

Example 10.1.6. .

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Not a polynomial: the exponent is not an integer.

Example 10.1.7. .

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, so it is not a polynomial (negative exponent).

Example 10.1.8. .

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A polynomial, already in order :)

Example 10.1.9. .

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so it is not a polynomial.

Example 10.1.10. .

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, which is a polynomial.

Example 10.1.11. .

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, so it is not a polynomial.

Example 10.1.12. .

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which is a polynomial.

Working with coefficients

Problem 10.1.13. Suppose . Find .

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So , since . Additionally, .

Expanding the first few powers: The sums of the coefficients of go so

Oh wait: the sum of the coefficients is the value at , which is :/

Example 10.1.14. for all . Find .

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so , , , . Therefore

Exercises

Exercise 10.1.15. Find the degree of , given that and are polynomials such that and , with .

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Let and , with . Then the leading term is given by WLOG let all with be , and , so that Ah, so now we see we have a leading term of , therefore of degree . Thus