Question

By Actual (Long) Division, find the Quotient & Remainder when -
${x}^{3} - 6 {x}^{2} + 2x - 4 \: is \: divided \: by \: 1 - \frac{3}{2} x$
CLUE -
$p(x) = x {}^{3} - 6 {x}^{2} + 2x - 4 \\ g(x) = - \frac{3}{2} x + 1$
$remainder = - \frac{136}{27}$
(STEPS - MANDATORY; EXPLANATION - OPTIONAL)
(Can also be as An Attachment)

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Answer 1

Answer:

Quotient : -2x² + 12x -4

Remainder : 8

Step-by-step explanation:

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The answer is in the attachment below ↓↓.

2bc4315a31be9bdddaaba93304b84699.pdf

Answer 2

Answer:

$\text{Quotient}=\frac{-1}{6}+\frac{16}{18}x^2+\frac{14}{54}\\\;\\\text{Reamainder}=-\frac{136}{27}$

Step-by-step explanation:

To find Quotient & reminder, we have used Long division method which is provided in attachment.

Since we have divided the quotient by g(x)/2 hence, quotient will be Q/2 of actual division while remainder will remain same.

Thus quotient will be

$\text{Quotient}=\frac{-1}{6}+\frac{16}{18}x^2+\frac{14}{54}$

Here is an alternate method to find Remainder:-

given

p(x) = x³ - 6x² + 2x - 4

g(x) = 1 - (3/2)x

Now putting 1 - (3/2)x = 0  

              ⇒   (3/2)x = 1

              ⇒    x = 2/3  

in p(x)

Remainder = (2/3)³ - 6(2/3)² + 2(2/3) - 4

                  = 8/27 - 24/9 + 4/3 - 4

                  = 8/27 - 8/3 + 4/3 - 4

                  = 8/27 - 4/3 - 4

                  = 8/27 - 36/27 - 4

                  = - 28/27 - 4

                  = -136/27

Note:- p(x) = Q * G(x)  + R

p(x) = Q * g(x)/2 + R

p(x) = Q/2 * g(x) + R

p(x) = (Q/2) * g(x) + R

Let Q' = Q/2

p(x) = Q' * g(x) + R  

Here Q' is new quotient which is equal to Q/2.