Question

pls answer this question fastly pls ​

Answer 1

Answer:

21/8

Step-By-Step-Explaination

Given :

p(x) = x³ - 6x² + 14x - 3

g(x) = 1 - 2x

• Step 1.

Find the root of g(x)

$\bf \: 1 - 2x = 0$

$\bf \implies \: x = \dfrac{1}{2}$

Remainder theorem

$\implies\bf \: p( \frac{1}{2} ) = {( \frac{1}{2} )}^{3} - 6 { (\frac{1}{2} )}^{2} + 14( \frac{1}{2} ) - 3$

$\implies \bf \: p( \frac{1}{2} ) = \frac{1}{8} - 6( \frac{1}{4} ) + 14( \frac{1}{2} ) - 3$

$\implies \bf \: p( \frac{1}{2} ) = \frac{1}{8} - \frac{ 3 \red{ \times 4}}{2 \red{ \times 4}} + \frac{7 \red{ \times 8}}{1 \red{ \times 8}} - \frac{ - 3 \red{ \times 8}}{1 \red { \times 8}}$

$\implies \bf \: p( \frac{1}{2} ) = \frac{1 - 12 + 56 - 24}{8}$

$\implies \bf \: p( \frac{1}{2} ) = \frac{21}{8}$

*Proving by long division method is done in attachment kindly refer to it.*

What is Remainder theorem ?

Remainder theorem just says that if a polynomial p(x) is divided by the polynomial x - a then the remainder will be p(a).

Dividend = (Divisor × Quotient) + Remainder [ That's what a remainder theorem is.]

Here , p(x) is a dividend .

and (x - a) = divisor.

Ploughing remainder theorem :

p(x) = (x-a)·q(x) + r

p(a) = (a-a)·q(a) + r

p(a) = (0)·q(a) + r

p(a) = r

Yes , p(a) is remainder ! and that's how it's known as remainder theorem.