Chapter 4.1, Problem 47E

To determine

To give an expression for a monomial and a polynomial.

Explanation of Solution

Given Information:To provide an example for a monomial and a polynomial.

Formula used: A monomial is an expression in algebra that contains one term, like $3xy$ . Monomials include: numbers, whole numbers and variables that are multiplied together, and variables that are multiplied together. A polynomial is a sum of monomials where each monomial is called a term.

Proof:

Any number, all by itself, is a monomial, like 5 or 2700. A monomial can also be a variable, like "m" or "b". It can also be a combination of these, like 98b or 78xyz. It cannot have a fractional or negative exponent. These are not monomials: 45x+3, 10 - 2a, or 67a - 19b + c.

Two rules about monomials are:

• A monomial multiplied by a monomial is also a monomial.
• A monomial multiplied by a constant is also a monomial.

When looking at examples of monomials, you need to understand different kinds of polynomials. Following is an explanation of polynomials, binomials, trinomials, and degrees of a polynomial.

Polynomials

A polynomial is an algebraic expression with a finite number of terms. These terms are in the form "axn" where "a" is a real number, "x" means to multiply, and "n" is a non-negative integer. Examples are 7a2 + 18a - 2, 4m2, 2x5 + 17x3 - 9x + 93, 5a-12, and 1273.

Binomials

A binomial is a polynomial with two terms. 3x + 1, 2x3 - 5x, x4 - 4, x - 19 are examples of binomials.

Trinomials

A trinomial is a polynomial with three terms. Examples of trinomials are 2x2 + 4x - 11, 4x3 - 13x + 9, 7x3 - 22x2 + 24x, and 5x6 - 17x2 + 97.