Consider-the-polynomial-P-x-x-10-6x-9-11x-8-3x-2-18x-7-Calculate-P-3-2-5-

Question Number 154959 by mathdanisur last updated on 23/Sep/21

Consider the polynomial

P (x) = x^{10} - {6 x}^{9} - {11 x}^{8} + {3 x}^{2} - 18 x - 7

Calculate : P (3 + 2 \sqrt{5})

Commented by prakash jain last updated on 23/Sep/21

x^{10} - 6 x^{9} - 11 x^{8} + 3 x^{2} - 18 x - 7

= x^{8} (x^{2} - 6 x - 11) + 3 x^{2} - 18 x - 7

= x^{8} [{(x - 3)}^{2} - 20] + 3 x^{2} - 18 x + 27 - 34

= x^{8} (0) + 3 {(x - 3)}^{2} - 34

= 3 \times 20 - 34

= 26

Commented by mathdanisur last updated on 23/Sep/21

Thankyou S er, very nice

Answered by Rasheed.Sindhi last updated on 23/Sep/21

P (x) = x^{10} - {6 x}^{9} - {11 x}^{8} + {3 x}^{2} - 18 x - 7; P (3 + 2 \sqrt{5}) = ?

= x^{10} - {6 x}^{9} - {11 x}^{8} + {3 x}^{2} - 18 x - 33 + 26

= x^{8} (x^{2} - 6 x - 11) + 3 (x^{2} - 6 x - 11) + 26

= (\underset{P_{1} (x)}{\underset{⏟}{x^{2} - 6 x - 11}}) (x^{8} + 3) + 26

P (x) = P_{1} (x) . (x^{8} + 3) + 26

P_{1} (x) = {(3 + 2 \sqrt{5})}^{2} - 6 (3 + 2 \sqrt{5}) - 11

= 9 + 20 + 12 \sqrt{5} - 18 - 12 \sqrt{5} - 11 = 0

P (3 + 2 \sqrt{5}) = (0) ({(3 + 2 \sqrt{5})}^{8} + 3) + 26 = 26

P (3 + 2 \sqrt{5}) = 26

Commented by mathdanisur last updated on 23/Sep/21

Very nice solution S er, thankyou

Answered by Rasheed.Sindhi last updated on 24/Sep/21

P (x) = x^{10} - {6 x}^{9} - {11 x}^{8} + {3 x}^{2} - 18 x - 7

\underline{P (3 + 2 \sqrt{5}) = ?}

BY S y n t h e t i c D i v i s i o n

∙ I f P (x) i s d i v i d e d b y x - k t h e n

r e m a i n d e r = P (k)

∙ k i s c a l l e d m u l t i p l y e r a n d h e r e

k = 3 + 2 \sqrt{5}

[\begin{matrix} \underset{⌣}{3 + 2 \sqrt{5}} \\ 1 & 1 \\ - 6 & 3 + 2 \sqrt{5} & - 3 + 2 \sqrt{5} \\ - 11 & 11 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 3 & 0 & 3 \\ - 18 & 9 + 6 \sqrt{5} & - 9 + 6 \sqrt{5} \\ - 7 & 33 & 26 \end{matrix}]

∴ P (3 + 2 \sqrt{5}) = 26

Commented by mathdanisur last updated on 24/Sep/21

Yes S er, cool, thank you