Q-1-p-x-3x-3-4x-2-5x-k-and-x-1-is-a-factor-of-p-x-find-the-value-of-k-and-the-remaining-two-factors-Q-2-Evaluate-r-1-3-2-r-

Question Number 35602 by Rio Mike last updated on 20/May/18

Q_{1} . p (x) = {3 x}^{3} + {4 x}^{2} + 5 x - k and

(x - 1) is a factor of p (x) find the

value of k and the remaining

two factors .

Q_{2} . E v a l u a t e \underset{r = 1}{\sum^{\infty}} 3^{2 - r} .

Answered by Rasheed.Sindhi last updated on 21/May/18

You can't use 'macro parameter character #' in math mode

If p (x) is divided by x - r then the

remainder is p (r) and if x - r a factor

of p (x) then this remainder must be

zero .

Here x - 1 a factor of p (x)

∴ p (1) = 0

Or 3 {(1)}^{3} + 4 {(1)}^{2} + 5 (1) - k = 0

k = 3 + 4 + 5 = 12

(\begin{matrix} 1) & 3 & 4 & 5 & - 12 \\ 3 & 7 & 12 \\ 3 & 7 & 12 & \overset{―}{∣ 0} \end{matrix})

Quotient = {3 x}^{2} + 7 x + 12

But {3 x}^{2} + 7 x + 12 is not factorable

∴ Factors : (x - 1) ({3 x}^{2} + 7 x + 12)

Answered by Rasheed.Sindhi last updated on 21/May/18

You can't use 'macro parameter character #' in math mode

3^{2 - r} is a GP

first term = a = 3^{2 - 1} = 3

common ratio = \frac{3^{2 - 2}}{3^{2 - 1}} = \frac{1}{3}

\underset{r = 1}{\sum^{\infty}} 3^{2 - r} = \frac{3}{1 - (1 / 3)} = 3 \times \frac{3}{2} = \frac{9}{2}