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Polynomial Definitions

Reference > Mathematics > Algebra > Polynomials
 

Having an understanding of terms from the previous page, we can move on to defining a polynomial and its related terminology.

Polynomial: A polynomial is one or more terms that contain only non-negative integer exponents, and which are combined with addition and subtraction. Note that the prefix "poly" means "many," so some will define a polynomial to requre at least two (or even three) terms. We do not follow that convention here, but consider any individual term to be a polynomial.

Examples: The following are examples of polynomials.
3, 3 + x, y5 - 1, 2x + 3y - 20z, x

Descending Order: A polynomial is written in descending order if its terms are arranged in order from largest degree to smallest degree. Note that this is the standard form for a polynomial; rearranging your polynomials to descending order should be automatic.

Examples: The following are polynomials written first in a random order, and then rearranged to descending order.
x3 + 7 - x5 in descending order is: -x5 + x3 + 7
xy5 - xy + x + x3y6 in descending order is: x3y6 + xy5 - xy + x

Note that for multivariable polynomials (like the example above) writing the polynomial in descending order may be made challenging by the fact that multiple terms may have the same degree. For example: x2 + 4y+ 4xy. In this case each term has degree two. We choose then, to write it in "descending order of x" or "descending order of y," which means we're treating the other variable as part of the coefficient, and ignore its exponent. Thus, x2 + 4y+ 4xy can be written in descending order of x as follows: x2 + 4xy + 4y

Degree of a Polynomial: The degree of a polynomial is the largest degree of any of its individual terms. If the polynomial is written in descending order, that will be the degree of the first term.

Examples: The following are examples of polynomials, with degree stated.
x+ 2x + 1 has degree 3.
x5y + x3y2 + xy3 has degree 6.

Note that with multivariable polynomials like the one above, we may choose to refer to it as a "polynomial in x" or a "polynomial in y." In this case, we would treat the other variable as part of the coefficient, and would ignore its exponents when determining the degree of the polynomial. For example, as a polynomial in x, x5y + x3y2 + xy3 has degree 5. As a polynomial in y, it needs to be rewritten in descending order of y: xy3 + x3y2 + x5y. The degree is 3.

Leading Coefficient: When written in descending order, the leading coefficient is the coefficient of the first term. Alternately, we can say that the leading coefficient is the coefficient of the term with highest degree.

Constant Term: A constant term is a term which contains no variable. If all terms contain a variable, then we say that the constant term is 0, because we can tack "+ 0" onto the polynomial without changing its meaning.

Examples: The following are polynomials with leading coefficients and constant terms labeled:
10x5 + 7x3 + 3 has leading coefficient 10 and constant term 3.
3x3 + x7 has leading coefficient 1 and constant term 0.

Special Names for Polynomials: Polynomials can have special names based on their number of terms, or the polynomial degree.

If a polynomial has one term, it can be called a monomial. If a polynomial has two terms it can be called a binomial. If a polynomial has three terms it can be called a trinomial.

If a polynomial's degree is 2, it can be called a quadratic. If its degree is 3 it can be called a cubic. If its degree is 4 it can be called a quartic.

Questions

1.
Why can't 5x-2 be a term in a polynomial?
2.
Why can't 3x5/2 be a term in a polynomial?
3.
Put the following polynomial in descending order: 3x + 2x5 - 4x2 + 2
4.
Put the following polynomial in descending order: 2xy - 2x2y + 3x2y3 - x2y2
5.
What is the degree of the following polynomial? x2 - 3x7 + 2x + 1 - 5x3
6.
What is the degree of the following polynomial? 5x2y3 + 2x3y - 3xy
7.
What is the leading coefficient in the example above?
8.
What is the constant in the example above?
9.
State the leading coefficient and constant term for the following polynomial: 6x2 + 3x + 1
10.
Give examples of: monomial; binomial; trinomial.
11.
Give examples of: quadratic; cubic; quartic.
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Terms, Coefficients, DegreeTerms, Coefficients, Degree
Evaluating PolynomialsEvaluating Polynomials
 

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