Question

Find the quotient and remainder
( 4x3+6x2-23x+18) ÷ (x+3)​

Answer 1

Given:

• $\sf{p(x) = 4x ^3+6x^2 -23x+18}$
• $\sf{g(x) = x+3}$

To Find :

• Reminder.

Solution :-

By Remainder Theorem :

Let g(x) = 0.

$: \implies{ \sf{x + 3 = 0}}$

$: \implies{ \sf{x = 0 - 3}}$

$: \implies{ \tt{x = - 3}}$

__________________

$\sf{p(x) = 4x^3+6x^2 -23x+18}$

$\sf{p(- 3) = 4(- 3)^3+6(-3)^2-23( - 3)+18}$

$\sf{p( - 3) = 4 ( - 27 )+6 (9) -23 ( - 3)+18}$

$\sf{p( - 3) = ( - 108)+54 + 69+18}$

$\sf{p( - 3) = - 108+54 + 69+18}$

$\sf{p( - 3) = - 108+141}$

$\to{\underline{ \boxed{\tt{p( - 3) = 33}}}}$

$\therefore{\underline {\pink{\sf{Remainder= 33.}}}}$

★ Remainder theorem :- If p(x) is a polynomial and p(x) is divided by the linear polynomial x + a, Then the Remainder is p(-a).

Answer 2

Answer :-

Given :- f(x) = 4x³ - 3x² + 2x + 1 ;

g(x) = x² - 3x + 6.

To Find :- Quotient & Remainder.

Solution :-

___________________

x² - 3x + 6) 4x³ - 3x² + 2x + 1 (4x + 9

4x³- 12x² + 24x

- + -

_____________________

9x² - 22x + 1

9x² - 27x + 54

- + -

_____________________

5x - 53

☃$\sf{Hence}$

⚘$\bf{The \: Quotient = 4x + 9 ;}$

☃$\rm{ \: Remainder = 5x - 53.}</p><p>$

$\huge \rm \pink{Division \: Algorithm :-}</p><p>$

• Division Algorithm is a division in which degree of Dividend is compares with degree of Divisor to know the quotient.