Examples of Monomials and Polynomials

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.

chart showing monomial, binomial and trinomial terms chart showing monomial, binomial and trinomial terms

Identifying a Monomial

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.

Degrees of Polynomials

The degree of a polynomial is the highest power of the variable in that polynomial, as long as there is only one variable. The degree of the polynomial 7x3 - 4x2 + 2x + 9 is 3, because the highest power of the variable “x” is 3. In 18s12 - 41s5 + 27, the degree is 12. The degree of this polynomial 8z + 2008 is 1, because “z” is in the first power.

If a polynomial has more than one variable, then the degree of that monomial is the sum of the exponents of those variables. In this polynomial, 24xyz, the degree is 3 because the sum of degrees of x, y and z is 1 + 1 + 1 = 3. The degree of this polynomial 12x4y2z7 is 13 because 4 + 2 + 7 = 13. 14x3 + 27xy - y has the degree of 6 because 3 + 1 + 1 + 1 = 6. So basically, you add the exponents of the variables together.

Algebra Terms

When looking at examples of monomials, binomials, and trinomials, it can seem a little confusing at first. It is just a classification for different polynomials with different numbers of terms. A second degree polynomial is also called a “quadratic.” Examples are 4x2, x2 – 9, or 6x2 + 13x + c.

Just for fun, a third degree polynomial is called a “cubic”, a fourth degree is called a "quartic", and a fifth degree polynomial is called a "quintic.

You may wonder where the word “quadriatic” comes from, because the prefix “quad” usually stands for four. The word comes from the Latin word for “making square.” So, in this instance, “quad” refers to the four corners of a square, like when you multiply six feet by six feet, you get 36 square feet.

History of Algebra

Algebra is a branch of pure mathematics. Pure mathematics differs from other disciplines because it is not necessarily applied to any particular situation but the concepts and beauty of math is investigated. The Greeks created geometric algebra, representing sides of objects with lines and letters. In the 3rd century AD, Diophantus, who is called the “father of algebra”, wrote several books called Arithmetica. In them, he explained how to solve algebraic equations.

The word “algebra” actually comes from Arabic and means “restoration.” Algebra actually started with the Babylonians, who were advanced mathematicians, dealing with quadratic and linear equations. Other civilizations at this time were still solving problems geometrically.

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