Polynomials Theory

Polynomials Theory - Transcript

1 4 Polynomials
A polynomial is the result when we have the HYPERLINK http www mathwords com s sum htm sum or HYPERLINK http www mathwords com d difference htm difference of HYPERLINK http www mathwords com t term htm terms which have HYPERLINK http www mathwords com v variable htm variables raised to HYPERLINK http www mathwords com p positive number htm positive HYPERLINK http www mathwords com i integers htm integer HYPERLINK http www mathwords com p power htm powers and which have HYPERLINK http www mathwords com c coefficient htm coefficients
The following are all polynomials
5x3 2x2 x 13
x2y3 xy
The following ARE NOT polynomials




Note Even though the prefix poly means many we use the word polynomial to refer to polynomials with 1 term HYPERLINK http www mathwords com m monomial htm monomials 2 terms HYPERLINK http www mathwords com b binomial htm binomials 3 terms HYPERLINK http www mathwords com t trinomial htm trinomials etc
Standard form for a polynomial in one variable
anxn an 1xn 1 a2x2 a1x a0
This means we write the terms in descending powers of x

A HYPERLINK http www mathwords com p polynomial htm polynomial with one HYPERLINK http www mathwords com t term htm term The following are all monomials 5x3 8 and 4xy
the coefficient
the base
the power exponent
For a term with one HYPERLINK http www mathwords com v variable htm variable the degree is the variable s HYPERLINK http www mathwords com e exponent htm exponent With more than one variable the degree is the HYPERLINK http www mathwords com s sum htm sum of the exponents of the variables
Examples
Binomial
A HYPERLINK http www mathwords com p polynomial htm polynomial with two HYPERLINK http www mathwords com t term htm terms which are not HYPERLINK http www mathwords com l like terms htm like terms
The following are all binomials
2x 3 3x5 8x4
2ab 6a2b5
NOT A BINOMIAL

Trinomial
A HYPERLINK http www mathwords com p polynomial htm polynomial with three HYPERLINK http www mathwords com t term htm terms which are not HYPERLINK http www mathwords com l like terms htm like terms
The following are all trinomials
x2 2x 3 3x5 8x4 x3
a2b 13x c

Degree of a polynomial
The highest HYPERLINK http www mathwords com d degree term htm degree of any term in the HYPERLINK http www mathwords com p polynomial htm polynomial
Multiplying two binomials
FOIL Method
A technique for HYPERLINK http www mathwords com d distribute htm distributing two HYPERLINK http www mathwords com b binomial htm binomials
The letters FOIL stand for First Outer Inner Last
First means multiply the HYPERLINK http www mathwords com t term htm terms which occur first in each binomial
Then Outer means multiply the outermost terms in the product
Inner means multiply the innermost two terms
Last means multiply the terms which occur last in each binomial
Then HYPERLINK http www mathwords com s simplify htm simplify the HYPERLINK http www mathwords com p product htm products and combine any HYPERLINK http www mathwords com l like terms htm like terms which may occur

Example x 2 x 5 x x x 5 2 x 2 5
First Outer Inner Last
x2 7x 10


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