p-x-1-x-1-2x-1-3x-1-14x-1-15x-the-absolute-value-of-the-coefficient-of-the-x-2-in-the-expression-

Question Number 179777 by luciferit last updated on 02/Nov/22

p (x) = (1 - x) (1 + 2 x) (1 - 3 x) \dots . (1 + 14 x) (1 - 15 x)

t h e a b s o l u t e v a l u e o f t h e c o e f f i c i e n t o f t h e x^{2} i n t h e e x p r e s s i o n ?

Answered by mr W last updated on 02/Nov/22

p (x) = \underset{k = 1}{\prod^{15}} [1 + {(- 1)}^{k} k x]

c o e f . o f x^{2} :

\frac{1}{2} \underset{k = 1}{\sum^{15}} {{(- 1)}^{k} k \underset{\underset{r \neq k}{r = 1}}{\sum^{15}} {(- 1)}^{r} r}

= \frac{1}{2} \underset{k = 1}{\sum^{15}} {{(- 1)}^{k} k \underset{r = 1}{\sum^{15}} {(- 1)}^{r} r - {(- 1)}^{2 k} k^{2}}

= \frac{1}{2} {{(\underset{r = 1}{\sum^{15}} {(- 1)}^{r} r)}^{2} - \underset{k = 1}{\sum^{15}} k^{2}}

= \frac{1}{2} {{[2 + 4 + 8 + \dots + 14 - (1 + 3 + 5 + \dots + 15)]}^{2} - \frac{15 \times (15 + 1) \times (2 \times 15 + 1)}{6}}

= \frac{1}{2} {{[\frac{7 \times 16}{2} - \frac{8 \times 16}{2}]}^{2} - \frac{15 \times 16 \times 31}{6}}

= - 588

∣ c o e f . o f x^{2} ∣= 588

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