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Transcript
Math 3 Name:_________________________10/11 Dividing Polynomials First, let’s consider what happens when we divide numbers. Say we divide 13 by 5. 2 quotient 5 13 We could write this as 10 13 3 remainder 2 3 5 5 Another way of thinking about this example is: 13 2 5 3 Division of polynomials is similar. If we divide a polynomial f (x) by ( x r ) we obtain a result of the form: f ( x) ( x r )q(r ) R where q (r ) is the quotient and R is the remainder. Example 1: f ( x) 3x 2 5x 8 by ( x 2) . 3x 11 2 x 2 3x 5 x 8 3x 2 6 x 11x 8 11x 22 14 We can conclude that: 3x 2 5x 8 ( x 2)(3x 11) 14 where the quotient q( x) 3x 11 and the remainder R=14 We can also write our answer as: 14 3x 2 4 x 8) ( x 2) 3x 11 Dividing Polynomials x2 Synthetic Division This is another way of dividing a polynomial by a binomial. Say we were dividing f ( x) x 2 5 x 6 by ( x 1) First, write the coefficients ONLY inside an upside-down division symbol: Make sure you leave room inside, underneath the row of coefficients, to write another row of numbers later. Since we are dividing by ( x 1) , we look at when this expression is equal to 0. x 1 0 when x 1. Put the this number, x = 1, at the left: Take the first number inside, representing the leading coefficient, and carry it down, unchanged, to below the division symbol: Multiply this carry-down value by the test zero, and carry the result up into the next column: Add down the column: Multiply the previous carry-down value by the test zero, and carry the new result up into the last column: Add down the column: This last carry-down value is the remainder. Remainder Theorem If a polynomial f(x) is divided by (x-r) and a remainder R is obtained, then f(r) = R. Example 2: Use the remainder theorem to find the remainder for Example 1, which was f ( x) 3x 2 5x 8 by ( x 2) . Since we are dividing f ( x) 3x 2 5x 8 by ( x 2) , we let x = 2 because this is when x 2 0 . Hence the remainder, R, is given by: R f (2) 3(2) 2 5(2) 8 14 This is the same remainder we achieved before. Factor Theorem If the remainder f(r) = R = 0, then (x-r) is a factor of f(x). The Factor Theorem is powerful because it can be used to find the roots of polynomial equations. Example 3: Is x 4 a factor of 3x 3 x 2 20 x 5 ? For this question we need to find out if dividing 3x 3 x 2 20 x 5 by x 4 leaves a remainder. If the remainder is 0, then x 4 is a factor of 3x 3 x 2 20 x 5 . We can find this easily by using the Remainder Theorem. Since x 4 0 when x 4 , we find what f (4) is: f (4) 2(4) 3 (4) 2 20(4) 5 123 The remainder 0 , so x 4 is not a factor of 3x 3 x 2 20 x 5 . Example 4: Is x 1 a factor of f ( x) x 3 2 x 2 5x 6 In this case we need to test the remainder r = -1. R f (r ) f (1) (1) 3 2(1) 2 5(1) 6 1 2 5 6 0 Therefore, since f (1) 0 , we conclude that x 1 is a factor of f ( x) x 3 2 x 2 5x 6