GED Mathematical Reasoning: Introduction To Polynomials

  • The prefix “poly” means “many” so it makes sense that a polynomial is a string of one or more terms being added or subtracted.
  • A polynomial may contain many more than three terms, but we only have special names for those having one, two or three terms.

A polynomial containing only one term is called a “monomial.”  For example: 4x^2 and -12

A polynomial containing two terms is called a “binomial.”  For example:  3a -9 and 16m^3 +5n^2

A polynomial containing three terms is called a “trinomial.”  For example:  7x^2 +x - 10

  • The number in front of the variable part  is called “coefficient” and every term contains a coefficient.

consider 7x^2 + x - 10

The coefficient of the first term 7x^2 is 7

The coefficient of the second term x is 1

The coefficient of the third term, even though there appears to be no variable part, is -10

  • When it comes to polynomials, the sign preceding a term goes with the term. What I mean is: If the sign is addition, the term is positive. If the sign is subtraction, the term is negative.
  • A term that appears to have no variable part is called a “constant” term since there is no variable to make it vary – meaning it will always remain the same. For example, -10 in 7x^2 + x - 10
  • The degree of a polynomial containing only one variable is equal to the largest exponent.

 

Example 1

Identify each term of the polynomial: 2a^2 - 3ab + \frac{4}{5}b^2

 

The polynomial in example one consists of three terms:

2a^2, -3ab and \frac{4}{5}b^2

Notice that we state the middle term to be negative, since there is a subtraction sign preceding 3ab. This is because we can rewrite subtraction using the idea of “adding the opposite,” which makes the term negative.

2a^2 - 3ab + \frac{4}{5}b^2 = 2a^2 +  -3ab + \frac{4}{5}b^2

 

Example 2

Classify the polynomial in Example 1 as a monomial, binomial or trinomial.

 

Since the polynomial in example one has three terms, it is classified as a trinomial.

 

Example 3

State the degree of the polynomial: -14x^3 - 10x^2 + 9x - 5

 

This polynomial contains 4 terms: -14 “x” cubed, -10 “x” squared, 9 “x” and -5. The coefficients in this polynomial are:  -14, -10, 9, and -5. And the constant term is:  -5.

The degree of a polynomial containing only one variable is equal to the largest exponent. The polynomial shown in example 3 involves only one variable, which is “x,” and the largest power of “x” is 3. Therefore, the degree of the polynomial is 3.

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