Explanation Given information: ( x − r ) ( x − s ) ( x − t ) = x 3 − ( r + s + t ) x 2 + ( r s + r s + s t ) x − r s t . Calculation: ( x − r ) ( x − s ) ( x − t ) r and s and t are odd integers ( x − r ) ( x − s ) ( x − t ) = x 3 − ( r + s + t ) x 3 + ( r s + r t + s t ) x − r s t The coefficient of x 3 is 1, which is odd. Since the sum of two odd integers is even, r + s is even. Since the sum of an even and an odd integer is odd, r + s + t is odd. Since the product of two odd integers is odd, rs is odd, rt is odd and st is odd. Since the sum of two odd integers is even, rs + rt is even. Since the sum of an even and an odd integer is odd, rs + rt + st is odd. Since the product of two odd integers is odd, rs is odd and then rst is also odd. Thus the coefficients of the polynomial x 3 − ( r + s + t ) x 3 + ( r s + r t + s t ) x − r s t are odd ( x 3 ) , odd ( x 2 ) , odd ( x ) and odd (constant). r and s and t are even integers ( x − r ) ( x − s ) ( x − t ) = x 3 − ( r + s + t ) x 3 + ( r s + r t + s t ) x − r s t The coefficient of x 3 is 1, which is odd. Since the sum of two even integers is even, r + s is even and thus r + s + t is also even. Since the product of two even integers is even, rs is even, rt is even and st is even. Since the sum of two even integers is even, rs + rt is even and thus rs + rt + st is also even. Since the product of two even integers is even, rs is even and then rst is also even. Thus the coefficients of the polynomial x 3 − ( r + s + t ) x 3 + ( r s + r t + s t ) x − r s t are odd ( x 3 ) , even ( x 2 ) , even ( x ) and even (constant). One integer of r, s and t is odd, others are even. ( x − r ) ( x − s ) ( x − t ) = x 3 − ( r + s + t ) x 3 + ( r s + r t + s t ) x − r s t The coefficient of x 3 is 1, which is odd To determine (b) To check: Whether x 3 + 7 x 2 − 8 x − 27 can be written as a product of three polynomials with integer coefficients or not.

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ISBN: 9780357035283

Author: EPP

Publisher: Cengage

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Chapter 4.3, Problem 34ES

To determine

(a)

To derive:

A result for cubic polynomials.

To determine

(b)

To check:

Whether x3+7x2−8x−27 can be written as a product of three polynomials with integer coefficients or not.

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Chapter 4.3, Problem 33ES

Chapter 4.3, Problem 35ES

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Chapter 4 Solutions

Discrete Mathematics With Applications

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(a) To derive: A result for cubic polynomials.

Solution Summary: The author explains that the coefficients of r, s, and t are all odd.