3.5 - 3.5: Upper and Lower Bounds for Roots The Upper and...

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3.5: More on Zeros of Polynomial Functions The Upper and Lower Bound Theorem helps us rule out many of a polynomial equation's possible rational roots. The Upper The Upper and and Lower Lower Bound Bound Theorem Theorem Let f ( x ) be a polynomial with real coefficients and a positive leading coefficient, and let a and b be nonzero real numbers. 1. 1. Divide f ( x ) by x - b (where b 0) using synthetic division. If the last row containing the quotient and remainder has no negative numbers, then b is an upper bound for the real roots of f ( x ) = 0. 2. 2. Divide f ( x ) by x - a (where a < 0) using synthetic division. If the last row containing the quotient and remainder has numbers that alternate in sign (zero entries count as positive or negative), then a is a lower bound for the real roots of f ( x ) = 0. Upper and Lower Bounds for Roots

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EXAMPLE: Finding Bounds for the Roots Show that all the real roots of the equation 8 x 3 + 10 x 2 - 39 x + 9 = 0 lie between –3 and 2. Solution We begin by showing that 2 is an upper bound. Divide the polynomial by x - 2. If all the numbers in the bottom row of the synthetic division are non-negative, then 2 is an upper bound . 2 8 10 - 39 9 16 52 26 8 26 13 35 All numbers in this row are nonnegative. 3.5: More on Zeros of Polynomial Functions more
EXAMPLE: Finding Bounds for the Roots Show that all the real roots of the equation 8 x 3 + 10 x 2 - 39 x + 9 = 0 lie between –3 and 2. Solution The nonnegative entries in the last row verify that 2 is an upper bound. Next, we show that - 3 is a lower bound. Divide the polynomial by x - ( - 3), or x + 3. If the numbers in the bottom row of the synthetic division alternate in sign, then - 3 is a lower bound. Remember that the number zero

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3.5 - 3.5: Upper and Lower Bounds for Roots The Upper and...

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