Lesson 5 - Transforming Polynomials Example 1: Find the...

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Transforming Polynomials Example 1: Find the polynomial whose roots are reciprocals ( 1 r ) of the roots of x 4 - 3 x 2 + x - 9. So f ( x ) = x 4 - 3 x 2 + x - 9. Then if r 1 , r 2 , r 3 , and r 4 are the roots of f , then f ( 1 x ) = 0 for 1 r 1 , 1 r 2 , 1 r 3 , and 1 r 4 . This is because: f ( 1 1 /r 1 ) = f ( r 1 ) = 0 So we have a function that has the roots we want, but it is not a poly- nomial. So we say g ( x ) = x 4 f ( 1 x ) which has the same roots as f ( 1 x ) and is a polynomial. g ( x ) = x n ( a n x n + a n - 1 x n - 1 + · · · + a 1 x + a 0 ) g ( x ) = a 0 x n + a 1 x n - 1 + · · · + a n g ( x ) = x 4 ( 1 x 4 - 3 x 2 + 1 x - 9) g ( x ) = 1 - 3 x 2 + x 3 - 9 x 4 So the polynomial we want is the original polynomial with coefficients reversed. Example 2: Find a polynomial whose roots are twice those of f ( x ) = x 4 - 3 x 2 + x - 9. Then if f ( r ) = 0, kr is a root of f ( x k ) (in this example k = 2). f
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Lesson 5 - Transforming Polynomials Example 1: Find the...

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