Algebra 1 HW 1 S

# Algebra 1 HW 1 S - (1 Prove that for any n 1 xn y n =(x...

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(1) Prove that for any n 1 , (1) x n y n =( x y )( x n 1 + x n 2 y + ··· + y n 1 ) . Solution : The proof will be by induction on n . One convenient way to do it, which simpliFes the algebra involved, is to Frst show that for any n 1, (2) 1 t n =(1 t )(1 + t + ··· + t n 1 ) . Then setting t = y/x we get (1) after multiplication by x n . So let’s prove (2). ±or n =1the equality (2) is clearly true. Assume now (induction hypothesis) that it holds for n = k , so that (3) 1 t k =(1 t )(1 + t + ··· + t k 1 ) , and let’s try to show that (2) also holds for n = k + 1. And indeed, 1+ t + ··· + t k 1 + t k =(1+ t + ··· + t k 1 )+ t k = 1 t k 1 t + t k (by the induction hypothesis (3)) = 1 t k +( t k t k +1 ) 1 t = 1 t k +1 1 t , as was to be shown ° . (2) Let a and b be nonzero integers such that b = aq + r , where 0 r<a . Prove that ( b, a )=( a, r ) . [As will be clear from the proof, the assumption on r is not needed here.] Solution : ±irst we show that (4) ( b, a )d iv ides( a, r ): indeed, by deFnition ( b, a ) divides both b

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Algebra 1 HW 1 S - (1 Prove that for any n 1 xn y n =(x...

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