if-p-x-2-2p-x-x-2-5x-3-find-p-x-

Question Number 129489 by mr W last updated on 16/Jan/21

i f p (x + 2) - 2 p (x) = x^{2} - 5 x - 3

f i n d p (x)

Answered by bramlexs22 last updated on 16/Jan/21

let p (x) = {a x}^{2} + b x + c

p (x + 2) = a {(x + 2)}^{2} + b (x + 2) + c

= {a x}^{2} + (4 a + b) x + (4 a + 2 b + c)

2 p (x) = 2 {a x}^{2} + 2 b x + 2 c

p (x + 2) - 2 p (x) = - {a x}^{2} + (4 a - b) x + (4 a + 2 b - c)

- {a x}^{2} + (4 a - b) x + (4 a + 2 b - c) = x^{2} - 5 x - 3

{\begin{cases} a = - 1 \\ - 4 - b = - 5; b = 1 \\ - 4 + 2 - c = - 3; c = 1 \end{cases}

p (x) = - x^{2} + x + 1

Commented by mr W last updated on 16/Jan/21

t h a n k s s i r!

b u t i s t h i s s o l u t i o n u n i q u e ? a r e t h e r e

a n y o t h e r p o s s i b l i t i e s ?

Answered by liberty last updated on 16/Jan/21

let p (x + 2) = u_{n + 2} and p (x) = u_{n}

homogenous equation λ^{2} - 2 = 0; λ = \pm \sqrt{2}

u_{n} = C_{1} {(\sqrt{2})}^{n} + C_{2} {(- \sqrt{2})}^{n} = λ {(\sqrt{2})}^{n}

particular solution : {ax}^{2} + bx + c \to {\begin{cases} u_{n + 1} = 2 ax + b \\ u_{n + 2} = 2 a \end{cases}

comparing : 2 a - ({ax}^{2} + bx + c) = x^{2} - 5 x - 3

- {ax}^{2} - bx + 2 a - c = x^{2} - 5 x - 3

{\begin{cases} a = - 1; b = 5 \\ - 2 - c = - 3; c = 1 \end{cases}

∴ p (x) = λ {(\sqrt{2})}^{x} - x^{2} + 5 x + 1

Commented by mr W last updated on 16/Jan/21

t h a n k s!

Commented by liberty last updated on 16/Jan/21

yes \dots i meant like that .

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