Name: Derivative of Logarithm and Exponentials
Uploaded: 2019-07-19
Duration: 42 min 29 s
Description: In this video, we will learn the concept of finding the derivative of logarithamic functions and also exponential functions with examples in detail

Question

Find integers

' a'

and

' b'

such that

(x^{2} - x - 1)

divides

a x^{17} + b x^{16} + 1

Accepted Answer

Let

$f (x) = a x^{17} + b x^{16} + 1$

Since, $(x^{2} - x - 1)$ divides $f (x)$ , therefore,

$f (x) = (x^{2} - x - 1) P (x)$

where, $P (x)$ is the quotient.

Let

$P (x) = c_{1} x^{15} + c_{2} x^{14} + . . . . + c_{15} x + c_{16}$

Then

$f (x) = (x^{2} - x - 1) (c_{1} x^{15} + c_{2} x^{14} + . . . . + c_{15} x + c_{16})$

$\Rightarrow a x^{17} + b x^{16} + 1 = (x^{2} - x - 1) (c_{1} x^{15} + c_{2} x^{14} + . . . . + c_{15} x + c_{16})$

Put $x = 0$ , then we get

$1 = (- 1) (c_{16}) \Rightarrow c_{16} = - 1$

Now, comparing coefficients of $x$ , we get

$- c_{16} - c_{15} = 0 \Rightarrow c_{15} = 1$

Now, comparing coefficients of $x^{2}$ , we get

$- c_{14} - c_{15} + c_{16} = 0$

$\Rightarrow c_{14} = - 2$

Now, comparing coefficients of $x^{3}$ , we get

$c_{15} - c_{14} - c_{13} = 0 \Rightarrow c_{13} = 3$

Now, comparing coefficients of $x^{4}$ , we get

$c_{14} - c_{13} - c_{12} = 0 \Rightarrow c_{12} = - 5$

Now, comparing coefficients of $x^{5}$ , we get

$c_{13} - c_{12} - c_{11} = 0 \Rightarrow c_{11} = 8$

Now, comparing coefficients of $x^{6}$ , we get

$c_{12} - c_{11} - c_{10} = 0 \Rightarrow c_{10} = - 13$

Now, comparing coefficients of $x^{7}$ , we get

$c_{11} - c_{10} - c_{9} = 0 \Rightarrow c_{9} = 21$

Now, comparing coefficients of $x^{8}$ , we get

$c_{10} - c_{9} - c_{8} = 0 \Rightarrow c_{8} = - 34$

Now, comparing coefficients of $x^{9}$ , we get

$c_{9} - c_{8} - c_{7} = 0 \Rightarrow c_{7} = 55$

Now, comparing coefficients of $x^{10}$ , we get

$c_{8} - c_{7} - c_{6} = 0 \Rightarrow c_{6} = - 89$

Now, comparing coefficients of $x^{11}$ , we get

$c_{7} - c_{6} - c_{5} = 0 \Rightarrow c_{5} = 144$

Now, comparing coefficients of $x^{12}$ , we get

$c_{6} - c_{5} - c_{4} = 0 \Rightarrow c_{4} = - 233$

Now, comparing coefficients of $x^{13}$ , we get

$c_{5} - c_{4} - c_{3} = 0 \Rightarrow c_{3} = 277$

Now, comparing coefficients of $x^{14}$ , we get

$c_{2} = - 510$

Now, comparing coefficients of $x^{15}$ , we get

$c_{1} = 787$

Now, comparing coefficients of $x^{16}$ , we get

$b = c_{2} - c_{1} = - 1297$

Now, comparing coefficients of $x^{17}$ , we get

$a = c_{1} = 787$

Find integers $' a'$ and $' b'$ such that $(x^{2} - x - 1)$ divides $a x^{17} + b x^{16} + 1$

Important Questions on Algebra

Learn and Prepare for any exam you want !

Find integers 'a' and 'b' such that x2-x-1 divides ax17+bx16+1

Important Questions on Algebra

Learn and Prepare for any exam you want !

Find integers $' a'$ and $' b'$ such that $(x^{2} - x - 1)$ divides $a x^{17} + b x^{16} + 1$