find the quotient and remainder on dividing 6x²-5x+1 by 2
Answers
Answer:
p(x)=x
3
−6x
2
+15x−8 ∴ degree of p(x) is 3.
g(x)=x−2 ∴ degree of g(x) is 1.
∴ degree of quotient q(x)=3−1=2 and degree of remainder r(x) is zero.
Let, q(x)=ax
2
+bx+c (Polynomial of degree 2) and r(x)=k (constant polynomial)
By using division algorithm, we have
p(x)=[g(x)×q(x)+r(x)]
=x
3
−6x
2
+15x−8=(x−2)(ax
2
+bx+c)+k
=ax
3
+bx
2
+cx−2ax
2
−2bx−2c+k
∴x
3
−6x
2
+15x−8=ax
3
+(b−2a)x
2
+(c−2b)x−2c+k
We have cubic polynomials on both the sides of the equation.
∴ Let us compare the coefficients of x
3
,x
2
,x and k to get the values of a,b,c.
1=a coefficient of x
3
on both sides
−6=b−2a coefficient of x
2
on both sides
15=c−2b coefficient of x on both sides
−8=−2c+k constant term on both sides
Let us solve these equations to get the values of b,c and k.
b−2a=−6∴b=−6+2a=−6+2(1)=−4
c−2b=15∴c=15+2b=15+2(−4)=7
−2c+k=−8∴k=−8+2c=−8+2(7)=6
q(x)=ax
2
+bx+c=(1)x
2
+(−4)x+7=x
2
−4x+7 and r(x)=k=6
∴ the quotient is x
2
−4x+7 and remainder is 6.
Step-by-step explanation:
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