Use the properties of even and odd integers that are listed in Example to do exercise.Exam...

Use the properties of even and odd integers that are listed in Example to do exercise.

Example

Deriving Additional Results about Even and Odd Integers

Suppose that you have already proved the following properties of even and odd integers:

1. The sum, product, and difference of any two even integers are even.

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2. The sum and difference of any two odd integers are even.

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3. The product of any two odd integers is odd.

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4. The product of any even integer and any odd integer is even.

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5. The sum of any odd integer and any even integer is odd.

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6. The difference of any odd integer minus any even integer is odd.

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7. The difference of any even integer minus any odd integer is odd.

Use the properties listed above to prove that if a is any even integer and b is any odd integer, then  is an integer.

Solution

Suppose a is any even integer and b is any odd integer. By property 3, b2 is odd, and by property 1, a2 is even. Then by property 5, a2 + b2 is odd, and because 1 is also odd, the sum (a2 + b2)+ 1 = a2 + b2 + 1 is even by property 2. Hence, by definition of even, there exists an integer k such that a2 + b2 + 1 = 2k. Dividing both sides by 2 gives , which is an integer. Thus  is an integer [as was to be shown].

Exercise

When expressions of the form (x −r)(x − s) are multiplied out, a quadratic polynomial is obtained. For instance, (x −2)(x −(−7))= (x −2)(x + 7) = x2 + 5x − 14.

a. What can be said about the coefficients of the polynomial obtained by multiplying out (x −r)(x − s) when both r and s are odd integers? when both r and s are even integers? when one of r and s is even and the other is odd?

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b. It follows from part (a) that x2 − 1253x + 255 cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.