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Zeros of Polynomial Functions (Part 2)

# Zeros of Polynomial Functions (Part 2) - The Upper and...

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The Upper and Lower Bound Theorem

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Upper Bound If you divide a polynomial function f ( x ) by ( x - c ), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f ( x ) = 0. Note that two things must occur for c to be an upper bound. One is c > 0 or positive. The other is that all the coefficients of the quotient as well as the remainder are positive. Lower Bound If you divide a polynomial function f ( x ) by ( x - c ), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f ( x ) = 0. Special note that zeros can be either positive or negative. Note that two things must occur for c to be a lower bound. One is c < 0 or negative. The other is that successive coefficients of the quotient and the remainder have alternating signs. Example 1 : Show that all real roots of the equation lie between - 4 and 4. In other words, we need to show that - 4 is a lower bound and 4 is an upper bound for real roots of the given equation. Checking the Lower Bound: Lets apply synthetic division with - 4 and see if we get alternating signs:
Note how c = -4 < 0 AND the successive signs in the bottom row of our synthetic division alternate . You know what that means? - 4 is a lower bound for the real roots of this equation. Checking the Upper Bound: Lets apply synthetic division with 4 and see if we get all positive: Note how c = 4 > 0 AND the all of the signs in the bottom row of our synthetic division are positive. You know what that means? 4 is an upper bound for the real roots of this equation. Since - 4 is a lower bound and 4 is an upper bound for the real roots of the equation, then that means all real roots of the equation lie between - 4 and 4. The Intermediate Value Theorem If f ( x ) is a polynomial function and f ( a ) and f ( b ) have different signs, then there is at least one value, c , between a and b such that f ( c ) = 0. In other words, when you have a polynomial function and one input value causes the function to be positive and the other negative, then there has to be at least one value in between them that causes the polynomial function to be 0. This works because 0 separates the positives from the negatives. So to go from positive to

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Zeros of Polynomial Functions (Part 2) - The Upper and...

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