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Discrete Mathematics with Graph Theory (3rd Edition) 156

# Discrete Mathematics with Graph Theory (3rd Edition) 156 -...

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154 Solutions to Exercises Therefore, f(x) + 10xf(x) + 25x 2 f(x) = ao + (al + lOao)x + (a2 + lOal + 25ao)x2 + ... + (an + lOan-l + 25an_2)X n + .... So f(x)(l + lOx + 25x 2 ) = 1 + 35x. Therefore, 1 f(x) = (1 + 5X)2 (1 + 35x) = (1 + 2( -5x) + ... + (n + 1)( -5xt + ... )(1 + 35x) = 1 + 25x - 275x 2 + ... + [(n + 1)( -5t + n( -5t- 1 (35)]xn + ... = 1 + 25x - 275x 2 + ... + [(1 - 6n)( _5)n]xn + ... and so an = (1 - 6n)( _5)n. 11. [BB] f(x) ao + alX + a2x2 + a3x3 + + anx n + + 2a n _lX n + + an_2 Xn + + 2an_3 Xn + 2xf(x) 2aox + 2alX2 + 2a2x3 + x 2 f(x) aox 2 + alx 3 + 2x 3 f(x 2aox3 + Therefore, f(x) - 2xf(x) - x 2 f(x) + 2x 3 f(x) = ao + (al - 2ao)x + (a2 - 2al - ao)x 2 + (a3 - 2a2 - al + 2ao)x3 + ... + (an - 2an -l - an -2 + 2an_3)X n + ... = 1 + x - x 2 since
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