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If-1-px-qx-2-8-1-8x-52x-2-kx-3-find-p-q-and-k-




Question Number 86684 by john santu last updated on 30/Mar/20
If (1+px+qx^2 )^8  = 1+8x+52x^2 +kx^3 +...  find p , q and k.
$$\mathrm{If}\:\left(\mathrm{1}+\mathrm{px}+\mathrm{qx}^{\mathrm{2}} \right)^{\mathrm{8}} \:=\:\mathrm{1}+\mathrm{8x}+\mathrm{52x}^{\mathrm{2}} +\mathrm{kx}^{\mathrm{3}} +… \\ $$$$\mathrm{find}\:\mathrm{p}\:,\:\mathrm{q}\:\mathrm{and}\:\mathrm{k}.\: \\ $$
Answered by jagoll last updated on 30/Mar/20
Commented by john santu last updated on 30/Mar/20
wow...super
$$\mathrm{wow}…\mathrm{super} \\ $$
Answered by Rio Michael last updated on 30/Mar/20
my try.    let px + qx = X   ⇒ (1 + px + qx^2 )^8  = (1 + X)^8  = 1 + 8x + 52x^2  + kx^3      (1+ X)^8  = 1 + 8X + ((8(7))/2) X^2  + ...                       = 1 + 8(px + qx^2 ) + 28(px + qx^2 )^2  +...                       = 1 + 8px + 8qx^2  + 28p^2 x^2  + 56pqx^3  + ...    as 1= 1    8p = 8 ⇒  p = 1   8q + 28p^2  = 52  ⇒ q = 3   k = 56pq +56 ⇒ k = 224
$$\mathrm{my}\:\mathrm{try}. \\ $$$$\:\:\mathrm{let}\:\mathrm{p}{x}\:+\:{qx}\:=\:{X} \\ $$$$\:\Rightarrow\:\left(\mathrm{1}\:+\:{px}\:+\:{qx}^{\mathrm{2}} \right)^{\mathrm{8}} \:=\:\left(\mathrm{1}\:+\:{X}\right)^{\mathrm{8}} \:=\:\mathrm{1}\:+\:\mathrm{8}{x}\:+\:\mathrm{52}{x}^{\mathrm{2}} \:+\:{kx}^{\mathrm{3}} \\ $$$$\:\:\:\left(\mathrm{1}+\:{X}\right)^{\mathrm{8}} \:=\:\mathrm{1}\:+\:\mathrm{8}{X}\:+\:\frac{\mathrm{8}\left(\mathrm{7}\right)}{\mathrm{2}}\:{X}^{\mathrm{2}} \:+\:… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{8}\left({px}\:+\:{qx}^{\mathrm{2}} \right)\:+\:\mathrm{28}\left({px}\:+\:{qx}^{\mathrm{2}} \right)^{\mathrm{2}} \:+… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{8}{px}\:+\:\mathrm{8}{qx}^{\mathrm{2}} \:+\:\mathrm{28}{p}^{\mathrm{2}} {x}^{\mathrm{2}} \:+\:\mathrm{56}{pqx}^{\mathrm{3}} \:+\:…\: \\ $$$$\:\mathrm{as}\:\mathrm{1}=\:\mathrm{1}\: \\ $$$$\:\mathrm{8}{p}\:=\:\mathrm{8}\:\Rightarrow\:\:{p}\:=\:\mathrm{1} \\ $$$$\:\mathrm{8}{q}\:+\:\mathrm{28}{p}^{\mathrm{2}} \:=\:\mathrm{52}\:\:\Rightarrow\:{q}\:=\:\mathrm{3} \\ $$$$\:{k}\:=\:\mathrm{56}{pq}\:+\mathrm{56}\:\Rightarrow\:{k}\:=\:\mathrm{224} \\ $$
Commented by jagoll last updated on 30/Mar/20
good..
$$\mathrm{good}.. \\ $$

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