Maths1020D_Week4_Sol.pdf - THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1020 General Mathematics(Fall 2017 uReply set 4 3 1 Let f(x

# Maths1020D_Week4_Sol.pdf - THE CHINESE UNIVERSITY OF HONG...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH1020 General Mathematics (Fall 2017) uReply set 4 1. Let f ( x ) = ln(2 x + 3) - 1 . The vertical asymptote of f ( x ) is - 3 2 . False. - 3 2 is a number, not a line. The vertical asymptote of f ( x ) should be x = - 3 2 . 2. The polynomial x 5 + x can be factorized as x 5 + x = x ( x 4 + 1) but not any further. False. Every polynomial can be factorized as a product of linear and irreducible quadratic polynomials. Indeed, x 5 + x = x ( x 4 + 1) = x ( x 2 + 2 x + 1)( x 2 - 2 x + 1) . 3. By partial fraction decomposition, there exists real constants A, B and C such that x 3 - 1 ( x 2 + 4)( x - 3) = Ax + B x 2 + 4 + C x - 3 . False. The rational function x 3 + 1 ( x 2 + 4)( x - 3) is improper. To decompose it, we should do long division to obtain a proper rational function first: x 3 - 1 ( x 2 + 4)( x - 3) = 1 + 3 x 2 - 4 x + 11 ( x 2 + 4)( x - 3) = 1 + x - 1 x 2 + 4 + 2 x - 3 . 4. By partial fraction decomposition, x 4 - x 2 + 1 x 3 ( x 2 - 1)( x 2 + x + 1) = A x 3 + B x 2 + C x + D x - 1 + E x + 1 + Fx + G x 2 + x + 1 for some constants A, B, C, D, E, F, G . True, as x 3 should be regarded as ( x - 0) 3 and x 2 + x + 1 can’t be factorized any further (Δ < 0). 5. The geometric series + X k =1 r k diverges for | r | ≥ 1. True. The series has partial sum