Question

If 2x cube + ax square - bx- 15 has 2x+3 as a factor and leaves a remainder-5 when divided by x-1 find the value of a and b

Answer 1

Let:
f(x)=x3+ax2+bx+6fx=x3+ax2+bx+6
(x−2) is a factor of f(x)=x3+ax2+bx+6.⇒f(2)=0⇒23+a×22+b×2+6=0⇒14+4a+2b=0⇒4a+2b=−14⇒2a+b=−7         ...(1)x-2 is a factor of fx=x3+ax2+bx+6.⇒f2=0⇒23+a×22+b×2+6=0⇒14+4a+2b=0⇒4a+2b=-14⇒2a+b=-7         ...1
Now, 
x−3=0⇒x=3x-3=0⇒x=3
By the factor theorem, we can say:
When f(x) will be divided by (x−3), 3 will be its remainder.⇒f(3)=3When fx will be divided by x-3, 3 will be its remainder.⇒f3=3

Now,
f(3)=33+a×32+b×3+6     =(27+9a+3b+6)     =33+9a+3bf3=33+a×32+b×3+6     =27+9a+3b+6     =33+9a+3b
Thus, we have:
    f(3)=3⇒33+9a+3b=3⇒9a+3b=−30⇒3a+b=−10      ...(2)    f3=3⇒33+9a+3b=3⇒9a+3b=-30⇒3a+b=-10      ...2
Subtracting 11 from 22, we get:
a = −-3
By putting the value of a in 11, we get the value of b, i.e., −-1.
∴ a = −-3 and b = −-1