5 7 64 89Check : Quotient * Divisor + R-(61 * 18) + 3 = 105Hence, the division isFind the quotient and the rema= 1101 = dividend2.EXERCISE 1G1. 504 - 12​

We know that the division algorithm states that:

Dividend=Divisor×Quotient+Remainder

It is given that the dividend is x

3

−6x

2

+11x−6, the divisor is x+2. And let the quotient be ax

2

+bx+c and the remainder be k. Therefore, using division algorithm we have:

x

3

−6x

2

+11x−6=(x+2)(ax

2

+bx+c)+k

⇒x

3

−6x

2

+11x−6=[x(ax

2

+bx+c)+2(ax

2

+bx+c)]+k

⇒x

3

−6x

2

+11x−6=(ax

3

+bx

2

+cx+2ax

2

+2bx+2c)+k

⇒x

3

−6x

2

+11x−6=ax

3

+(b+2a)x

2

+(c+2b)x+(2c+k)

By comparing the coefficients of the variables and the constant term we get:

a=1

b+2a=−6

⇒b+(2×1)=−6

⇒b+2=−6

⇒b=−6−2

⇒b=−8

c+2b=11

⇒c+(2×−8)=11

⇒c−16=11

⇒c=11+16

⇒c=27

2c+k=−6

⇒(2×27)+k=−6

⇒54+k=−6

⇒k=−6−54

⇒k=−60

Substituting the values, we get the quotient and remainder as follows:

q(x)=ax

2

+bx+c=x

2

−8x+27

r(x)=k=−60

Hence, the quotient is x

2

−8x+27 and the remainder is −60.