apply division algorithm to find quotient qx and remainder rx on dividing fx by gx
Answers
Answer:
Correct option is
C
Q=x
2
−8x+27 , R=−60
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.