If-partial-fraction-10x-2-px-18-2x-3-5x-2-x-2-can-be-written-as-q-2x-1-4-x-2-r-x-1-Then-find-the-value-of-p-q-2r-

Question Number 118485 by bramlexs22 last updated on 18/Oct/20

I f p a r t i a l f r a c t i o n

\frac{10 x^{2} + p x + 18}{2 x^{3} + 5 x^{2} + x - 2} c a n b e w r i t t e n

a s \frac{q}{2 x - 1} + \frac{4}{x + 2} + \frac{r}{x + 1} . T h e n f i n d t h e

v a l u e o f p - q + 2 r .

Answered by benjo_mathlover last updated on 18/Oct/20

\Rightarrow \frac{10 x^{2} + p x + 18}{2 x^{3} + 5 x^{2} + x - 2} \equiv \frac{q}{2 x - 1} + \frac{4}{x + 2} + \frac{r}{x + 1}

w e f i r s t f i n d t h e v a l u e o f p .

\Rightarrow 4 = {[\frac{10 x^{2} + p x + 18}{(2 x - 1) (x + 1)}]}_{x = - 2}

\Rightarrow 4 = [\frac{40 - 2 p + 18}{(- 1) (- 5)}] = \frac{58 - 2 p}{5}

\Rightarrow 20 = 58 - 2 p; p = 19

n o w w e f i n d t h e v a l u e o f q a n d r .

\Rightarrow q = {[\frac{10 x^{2} + 19 x + 18}{(x + 2) (x + 1)}]}_{x = \frac{1}{2}}

\Rightarrow q = [\frac{\frac{5}{2} + \frac{19}{2} + 18}{(\frac{5}{2}) (\frac{3}{2})}] = \frac{4 \times 30}{15} = 8

\Rightarrow r = {[\frac{10 x^{2} + 19 x + 18}{(2 x - 1) (x + 2)}]}_{x = - 1}

\Rightarrow r = [\frac{10 - 19 + 18}{(- 3) (1)}] = \frac{9}{- 3} = - 3

T h u s t h e v a l u e o f p - q + 2 r = 19 - 8 - 6 = 5

Answered by 1549442205PVT last updated on 18/Oct/20

\frac{10 x^{2} + p x + 18}{2 x^{3} + 5 x^{2} + x - 2} = \frac{q}{2 x - 1} + \frac{4}{x + 2} + \frac{r}{x + 1} .

\Leftrightarrow {10 x}^{2} + px + 18 = {qx}^{2} + 3 qx + 2 q + {8 x}^{2} + 4 x - 4 + {2 rx}^{2} + 3 rx - r

\Leftrightarrow {10 x}^{2} + px + 18 = (q + 2 r + 8) x^{2} + (3 q + 3 r + 4) x + 2 q - 2 r - 4

\Leftrightarrow {\begin{cases} q + 2 r + 8 = 10 (1) \\ 3 q + 3 r + 4 = p (2) \\ 2 q - 2 r - 4 = 18 (3) \end{cases}

Adding (3) to (1)

we get 3 q + 4 = 28 \Rightarrow q = 8, replace

into (1) we get r = - 3, replace into (2)

we get p = 3.8 - 3.3 + 4 = 19

Thus, (p, q, r) = (19, 8, - 3)

Therefore, p - q + 2 r = 5