Question

On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and – 2x + 4, respectively. Find g(x).

Answer 1

Dear Student.

We know that

Dividend= quotient*divisor+remainder                      (1)

Given,

Dividend =  x³ – 3x² + x + 2

Divisor = polynomial g(x),

the quotient and remainder were x – 2 and – 2x + 4, respectively.

Put this in equation(1)

x³ – 3x² + x + 2 =(x-2)*g(x)+(-2x+4)

x³ – 3x² + x + 2+2x-4 = (x-2)*g(x)

x³ – 3x²+3x-2 = (x-2)*g(x)

(x-2)( x²-x+1 )= (x-2)*g(x)

Hence, g(x)= ( x²-x+1 )

See the attachment.

Answer 2

Solution :

Given Dividend = x³-3x²+x+2 ,

Divisor = g(x) ,

Quotient = x - 2 ,

Remainder = -2x + 4

x-2)x³-3x²+x+2(x²-x+1
**** x³-2x²
___________
******-1x²+3x-2
******-x² + 2x
___________
*********** x - 2
*********** x - 2
___________
************ ( 0 )

Using the division algorithm,

Dividend=divisor×quotient+Remainder

x³-3x²+1x+2

= g(x) × ( x - 2 ) + ( -2x + 4 )

=> x³-3x²+3x-2 = g(x) × ( x - 2 )

=> g(x) = ( x³-3x²+3x-2 )/( x - 2 )

=> g(x) = x² - 1x + 1

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