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Chapter 2: Problem 95

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}+3 x^{2}-2 x+1\) (a) Upper: \(x=1\) (b) Lower: \(x=-4\)

Short Answer

Expert verified
Based on synthetic division, \(x=1\) is an upper bound for the real zeros of the given polynomial, while \(x=-4\) is not a lower bound.

Step by step solution

01

Test the Upper Bound

To test the upper bound, use synthetic division with \(x = 1\) as the divisor and the coefficients of the polynomial as the dividend. Set up the synthetic division table as follows:\n\n| 1 | 1 | 3 | -2 | 1 |\n|---------|-------|-----|------|----|\n| | 1 | 4 | 2 |\n\nAdd down the columns. The result will be the next row:\n\n| 1 | 1 | 4 | 2 |\n|----|-----|------|-----|\n| | 1 | 7 | 4 | 3 |\n\nThe last value in the final row, 3, is the remainder. Since this value is positive, we can conclude that \(x = 1\) is an upper bound for the real zeros.
02

Test the Lower Bound

To test the lower bound, use synthetic division with \(x = -4\) as the divisor. Set up the synthetic division table as follows:\n\n| -4 | 1 | 3 | -2 | 1 |\n|--------|-------|------|-----|-----|\n| | 1 | -1 | 2 |\n\nAdd down the columns. The result will be the next row:\n\n| -4 | 1 | -1 | 2 |\n|-----|-----| ------|-----|\n| | 1 | -3 | 10 | 9 |\n\nThe last value in the final row, 9, is the remainder. Since this value is positive, we can conclude that \(x = -4\) is not a lower bound for the real zeros.

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Most popular questions from this chapter

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-4 x^{3}+5 x^{2}-2 x-6\) (Hint: One factor is \(\left.x^{2}-2 x-2 .\right)\)

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}+36$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?
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