91Ó°ÊÓ

Every positive integer can be expressed as a sum of three or fewer perfect squares.

If \(c\) is a positive real number and \(x\) is any real number, then \(-c \leq x \leq c\) if, and only if, \(|x| \leq c\). (To prove a statement of the form " \(A\) if, and only if, \(B\)." you must prove "if \(A\) then \(B\) " and "if \(B\) then \(A . "\) )

For all real numbers \(x\) and \(y_{,}|x+y| \leq|x|+|y| .\) This result is called the triangle inequality. (Hint: Use 51 and 52 above.)

The difference of the squares of any two consecutive integers is odd.

For all nonnegative real numbers \(a\) and \(b, \sqrt{a b}=\sqrt{a} \sqrt{b}\). (Note that if \(x\) is a nonnegative real number, then there is a unique nonnegative real number \(y\), denoted \(\sqrt{x}\), such that \(\left.y^{2}=x_{0}\right)\)

If \(m\) and \(n\) are perfect squares, then \(m+n+2 \sqrt{m n}\) is also a perfect square. Why?

If \(p\) is a prime number, must \(2^{p}-1\) also be prime? Prove or give a counterexample.

When expressions of the form \((x-r)(x-s)\) are multiplied out, a quadratic polynomial is obtained. For instance, \((x-2)(x-(-7))=(x-2)(x+7)=x^{2}+5 x-14 .\) \(H\) 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? b. It follows from part (a) that \(x^{3}-1253 x+255\) cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.

Observe that \((x-r)(x-s)(x-t)\) $$ =x^{3}-(r+s+t) x^{2}+(r s+r t+s t) x-r s t . $$ a. Derive a result for cubic polynomials similar to the result in part (a) of exercise 60 for quadratic polynomials. b. Can \(15 x^{3}+7 x^{2}-8 x-27\) be written as a product of three polynomials with integer coefficients? Explain.