Also note that if both evaluations are positive or

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Unformatted text preview: rify that the roots of the following polynomial satisfy the rational root theorem. P ( x ) = 12 x 3 - 41x 2 - 38 x + 40 = ( x - 4 ) ( 3 x - 2 ) ( 4 x + 5) Solution From the factored form we can see that the zeroes are, x=4= 4 1 x= 2 3 x=- 5 4 Notice that we wrote the integer as a fraction to fit it into the theorem. Also, with the negative zero we can put the negative onto the numerator or denominator. It won’t matter. So, according to the rational root theorem the numerators of these fractions (with or without the minus sign on the third zero) must all be factors of 40 and the denominators must all be factors of 12. Here are several ways to factor 40 and 12. 40 = ( 4 ) (10 ) 40 = ( 2 ) ( 20 ) 40 = ( 5 ) ( 8 ) 12 = (1)(12 ) 12 = ( 3)( 4 ) 40 = ( -5 ) ( -8 ) 12 = ( -3)( -4 ) From these we can see that in fact the numerators are all factors of 40 and the denominators are all factors of 12. Also note that, as shown, we can put the minus sign on the third zero on either the numerator or the denominator and it will stil...
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