Page 34 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

1.91 If the sum of two consecutive natural numbers n and n +1 is the square of
another natural number m , show n is divisible by 4.

Proof: n + n +1 = m , i.e. m =2n +1. 2n +1 is an odd number, then m is also an
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2
2
odd number, then m has to be odd. Let m =2k +1 (k is a nonnegative integer).
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n = m −1 = (m−1)(m+1) =2k(k + 1). Since k(k + 1) is obviously an even number, then
2 2
n =2k(k + 1) is divisible by 4.

1.92 If x − 2x + ax − 6 and x +5x + bx +8 have a second order common factor,
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2
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2
determine the values of a, b .
Solution: Let

x − 2x + ax − 6= (x + px + q)(x + c) = x +(c + p)x +(cp + q)x + cq
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3
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2
x +5x + bx +8 = (x + px + q)(x + d)= x +(d + p)x +(dp + q)x + dq
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2
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Make the corresponding coefficients equal to have
p + c = −2, cp + q = a, cq = −6,d + p =5, dp + q =6, dq =8. From these six algebraic
equations, we obtain a = −1,b =6,c = −3,d =4,p =1,q =2.

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