The-expansion-of-1-px-qx-2-8-1-8x-52x-2-kx-3-What-are-the-values-of-p-q-and-k-

Question Number 148219 by iloveisrael last updated on 26/Jul/21

The expansion of {(1 + px + {qx}^{2})}^{8}

= 1 + 8 x + {52 x}^{2} + {kx}^{3} + \dots

What are the values of p, q and k

Answered by liberty last updated on 26/Jul/21

b y T r i n o m i a l t h e o r e m

{(1 + p x + {q x}^{2})}^{8} = \sum_{α + β + γ = 8} 1^{α} {(p x)}^{β} {({q x}^{2})}^{γ}

(i) x - t e r m = \frac{8!}{7! 1! 0!} {(p x)}^{1} = 8 p x

(i i) x^{2} - t e r m = \frac{8!}{6! 2! 0!} {(p x)}^{2} + \frac{8!}{7! 0! 1!} {({q x}^{2})}^{1} = (28 p^{2} + 8 q) x^{2}

(i i i) x^{3} - t e r m = \frac{8!}{5! 3! 0!} {(p x)}^{3} + \frac{8!}{6! 1! 1!} {(p x)}^{1} {({q x}^{2})}^{1}

= (56 p^{3} + 56 p q) x^{3}

c o m p a r i n g c o e f f i c i e n t s

{\begin{cases} 8 p = 8 \Rightarrow p = 1 \\ 28 p^{2} + 8 q = 52 \Rightarrow q = 3 \\ 56 p^{3} + 56 p q = k \Rightarrow k = 224 \end{cases}

Facebook Tweet Pin