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<< P.4. Radicals and Rational Exponents

P.5. Polynomials

Polynomials are a very important class of functions in mathematics. In order to define a polynomial, we must start with some other definitions. A * monomial* is a product of a real number, known as a

*, and a finite number of variables each raised to a positive integer power. Examples of monomials are 1, -4, x, 3x*

**coefficient**^{2}, -5xy, 2xyz, -x

^{2}y

^{2}, and πyz

^{3}. The

*is the sum of powers of all variables in the monomial. A*

**degree of a monomial***is a finite sum of monomials. The*

**polynomial***is the maximum degree of its monomials.*

**degree of a polynomial**The simplest type of polynomials are polynomials of one variable, more commonly known as univariate polynomials. A * univariate polynomial* is an expression of the form a

_{n}x

^{n }+ a

_{n-1}x

^{n-1 }+ ... + a

_{2}x

^{2}+ a

_{1}x + a

_{0}, where the the numbers a

_{0}, a

_{1}, a

_{2}, ..., a

_{n}are real valued constants known as the coefficients of the polynomial and n is a whole number known as the degree of the polynomial. Examples of univariate polynomials are x, 3x + 4, -4x

^{3}, x

^{2}+ 6x - 5, -4x

^{3}, and x

^{3 }- 3x

^{2 }+ 6x - 6. A

*is a polynomial of two or more variables. Some examples of multivariate polynomials are x + y, 3x + 5y - 6, 4xy, x*

**multivariate polynomial**^{2}+ xy - y

^{2}, xyz, and x

^{3 }- 3y

^{2 }+ 5xz - 8.

Polynomials of small degree have special names. The following table lists the names of polynomials of degree up to 9:

degree | name |
---|---|

0 |
constant |

1 |
linear |

2 |
quadratic |

3 |
cubic |

4 |
quartic |

5 |
quintic |

6 |
sextic |

7 |
septic |

8 |
octic |

9 |
nonic |

Table P.5.1: Names of Polynomials of Small Degree

**Example 1: **Determine the degrees and names of the following polynomials and identify them as univariate or multivariate.

- (a) x
- (b) 3x
^{8}+ 15x^{4 }+ 1 - (c) π + 3
- (d) π + x
- (e) 2x + 5y + 4
- (f) -8xyz
- (g) x
^{2}y^{2}+ 3xyz^{3}- 2z^{4}

**Solution:**

- (a) The polynomial x is univariate of degree 1 (linear) since x = x
^{1}. - (b) The polynomial 3x
^{8}+ 15x^{4 }+ 1 is univariate of degree 8 (octic) since 8 is the largest power of x in any of its monomials. - (c) The polynomial π + 3 is univariate of degree 0 (constant) since it does not contain any variables.
- (d) The polynomial π + x is univariate of degree 1 (linear) since 1 is the largest power of x of any of its monomials.
- (e) The polynomial 2x + 5y + 4 is multivariate of degree 1 (linear) since the monomials 2x and 5y each have maximal degree 1. (The remaining monomial, 4, has degree 0.)
- (f) The polynomial -8xyz is multivariate of degree 3 (cubic) since it is a monomial with the sum of powers of its variables equal to 3.
- (g) The polynomial x
^{2}y^{2 }+ 3xyz^{3}- 2z^{4}is multivariate of degree 5 (quintic), the monomial 3xyz^{3}having maximal degree 1 + 1 + 3 = 5.

A polynomial may be evaluated at particular values of its variables. We illustrate this with a couple of examples.

**Example 2: **Evaluate the polynomial 3x^{2 }+ 5x + 6 at x=2.

**Solution:** We substitue 2 for x and solve, obtaining 3*2^{2 }+ 5*2 + 6 = 3*4 + 5*2 + 6 = 12 + 10 + 6 = 28.

**Example 3: **Evaluate the polynomial x^{2 }+ 4xy - 5y^{2} at x=3; y=5.

**Solution: **Substituting these values we obtain 3^{2} + 4*3*5 - 5*5^{2} = 9 + 60 - 125 = -56.

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