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polynomials

# polynomials - Polynomials Basic concepts connected with...

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Polynomials Basic concepts connected with polynomials A polynomial in a single variable x 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 coefficients , , , , , a 0 a 1 a 2 . . . a - n 1 a n are real numbers, or, more generally, they can be complex numbers. The polynomial has degree n provided that a n 0. For example, + - π x 5 7 x 4 23 56 + - + x 3 2006 x 2 x 2 is a polynomial of degree 5. A polynomial such as the preceding example, defines an associated polynomial function f, where = ( ) f x + - π x 5 7 x 4 23 56 + - + x 3 2006 x 2 x 2. A zero of a polynomial ( ) P x is a real (or complex ) number x such that = ( ) P x 0. For example, the cubic polynomial = ( ) P x - - x 3 3 x 2 has the zeros - 1 and 2. In fact = ( ) P x ( ) - x 2 ( ) + x 1 2 , as can be seen as follows. = ( ) - x 2 ( ) + x 1 2 ( ) - x 2 ( ) + + x 2 2 x 1 = + + - - - x 3 2 x 2 x 2 x 2 4 x 2 = - - x 3 3 x 2. The zero = x - 1 is associated with the factor + x 1, which is appears with the power 2 in the factorisation of ( ) P x , and, as such, is said to have multiplicity 2. The following picture shows the graph of = y - - x 3 3 x 2. Note that it has the two x intercepts given by = x - 1 where the graph just touches the x axis but does not cross it, and = x 2, where the graph crosses the x axis. . For another example, the cubic polynomial = ( ) Q x + + x 3 3 x 2 has only one real number zero = x - 1 ( ) + 2 1 1 3 ( ) + 2 1 1 3 ~ - 0.596072,

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that is, = x - 1 r r , where = r 3 + 2 1. We can check that = x - 1 r r is a zero of ( ) Q x as follows ( but you can skip this calculation if you wish ). Making use of the formula = ( ) - a b 3 - + - a 3 3 a 2 b 3 a b 2 b 3 , we see that, when = x - 1 r r , = + + x 3 3 x 2 + + - 1 r r 3 3 - 1 r r 2 = - + - + - + 1 r 3 3 r 3 r r 3 3 r 3 r 2 = - + 1 r 3 r 3 2 = - + 1 + 2 1 ( ) + 2 1 2 = - + - 2 1 - 2 1 ( ) + 2 1 2 = - - - + 2 1 2 1 2 = 0. The polynomial = ( ) P x ( ) + x 5 3 ( ) - x 4 ( ) + x 1 2 = + + - - - - x 6 13 x 5 38 x 4 134 x 3 835 x 2 1175 x 500 has zeros = x - 5 (with multiplicity 3), = x 4 (with multiplicity 1), and = x - 1 (with multiplicity 2). The graph has x intercepts at these zeros. In the interval [ ] , 0 4 the graph extends beyond the picture and down to a minimum value of approximately -8250.16. In general, the graph will cross the x axis at a zero with odd multiplicity, but just touch the x axis without crossing at a zero with even multiplicity.
Example 1 : Find the zeros of the given polynomial = ( ) P x - + x 4 4 x 2 4, and state the multiplicity of each zero. Solution : = ( ) P x - + x 4 4 x 2 4 = ( ) - x 2 2 2 = [ ( ) + x 2 ( ) - x 2 ] 2 = ( ) + x 2 2 ( ) - x 2 2 . Hence ( ) P x has the two zeros - 2 and 2, each with multiplicity 2. Note : The following picture shows the graph of = y - + x 4 4 x 2 4. The graph just touches the x axis at the points ( ) , - 2 0 and ( ) , 2 0 . Since ( ) - x 2 2 2 is never negative, the graph never goes below the x axis and so never crosses the x axis. Example 2 : Find the zeros of the given polynomial = ( ) P x - ( ) - x 3 4 4 ( ) - x 3 2 , and state the multiplicity of each zero. Solution : = ( ) P x - ( ) - x 3 4 4 ( ) - x 3 2 = ( ) - x 3 2 [ ] - ( ) - x 3 2 4 = ( ) - x 3 2 [ ] ( ) - ( ) - x 3 2 ( ) + ( ) - x 3 2 = ( ) - x 3 2 ( ) - x 5 ( ) - x 1 .

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