Thus 3 x 5 23 x 4 34 x 3 38 x 2 64 x 24 x 2

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Unformatted text preview: 0 76 24 − 17 0 38 12 2 0 Thus, f ( 2 ) = 0 ⇒ x − 2 is a factor of f and 2 is a zero (root) of f. Thus, 3 x 5 − 23 x 4 + 34 x 3 + 38 x 2 − 64 x − 24 = ( x − 2 ) ( 3 x 4 − 17 x 3 + 38 x + 12 ) . Note that the remaining zeros of the polynomial f must also be zeros (roots) of this 4 3 first quotient polynomial q 1 ( x ) = 3 x − 17 x + 38 x + 12 . We will use this polynomial to find the remaining zeros (roots) of f, including another zero (root) of 2. 4 Trying 2 again: 3 Coefficien 17 ts of 3x − x + 38x + 12 3 − 17 0 38 12 6 3 − 22 − 44 − 12 − 11 − 22 −6 2 0 The remainder is 0. Thus, x − 2 is a factor of the quotient polynomial q 1 ( x ) = 3 x 4 − 17 x 3 + 38 x + 12 and 2 is a zero (root) of multiplicity of the polynomial f. Thus, 3 x 4 − 17 x 3 + 38 x + 12 = ( x − 2 ) ( 3 x 3 − 11 x 2 − 22 x − 6 ) Thus, 3 x 5 − 23 x 4 + 34 x 3 + 38 x 2 − 64 x − 24 = ( x − 2 ) 2 ( 3 x 3 − 11 x 2 − 22 x − 6 ) . Note that the remaining zeros of the polynomial q 1 (and f ) must also be zeros 3 2 (roots) of this...
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