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© 2017 The Authors. We analyse the clustering of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey Data Release 14 quasar sample (DR14Q).We measure the redshift space distortions using the power-spectrum monopole, quadrupole, and hexadecapole inferred from 148 659 quasars between redshifts 0.8 and 2.2, covering a total sky footprint of 2112.9 deg2. We constrain the logarithmic growth of structure times the amplitude of dark matter density fluctuations, fσ8, and the Alcock-Paczynski dilation scales that allow constraints to be placed on the angular diameter distance DA(z) and the Hubble H(z) parameter. At the effective redshift of zeff = 1.52, fσ8(zeff) = 0.420 ± 0.076, H(zeff) = [162 ± 12] (r sfid/rs) kms-1 Mpc-1, and DA(zeff) = [1.85 ± 0.11] × 103 (rs/r sfid) Mpc, where rs is the comoving sound horizon at the baryon drag epoch and the superscript 'fid' stands for its fiducial value. The errors take into account the full error budget, including systematics and statistical contributions. These results are in full agreement with the current Λ-Cold Dark Matter cosmological model inferred from Planck measurements. Finally, we compare our measurements with other eBOSS companion papers and find excellent agreement, demonstrating the consistency and complementarity of the different methods used for analysing the data.