Question

if x³+ax²+bx+6 has x-2, as a factor and leaves a remainder 3 when divided by x-3,find the values of a and b​

Answer 1

Step-by-step explanation:

f(x)=x³+ax²+bx+6

(x−2) in a factor ⇒f(2)=0

f(2)⇒2³+a(2)²+2b+6=0

⇒8+4a+2b+b=0

⇒4a+2b+14=0

⇒2a+b+7=0 -(1)

f(x−3)=3 (Remainder)

⇒f(3)⇒3³+a(3)²+b×3+6=3

⇒27+9a+3b+b=3

⇒9a+3b+30=0

⇒3a+b+10=0 _____ (2)

From (1) & (2)

b=−2a−7 & b=−10−3a

−2a−7=−10−3a

3a−2a=−10+7

a=−3

from (3)

b=−2(a)−7=2(−3)−7

=6−7

b=−1