Question

If (x-2) is a factor of the expression 2xcube+axsquare+bx-14 and when the expression is divided by (x-3), it leaves a remainder 52, find the values a and b.

Answer 1

hello...

p(x)= 2x³+ax²+bx-14
put (x-2) factor of p(x)
x-2=0
x=2
put x=2
2(2)³+a(2)²+b(2)-14=0
16+2a+2b-14=0
8+2a+2b-14=0
2a+b+1=0... eq 1

Now divide p(x) by x-3
(x-3)=0
x=3

Now put x=3
2(3)³+a(3)²+b(3)-14=52
2(27)+a(9)+b(3)-14=52
54+9a+3b-14=52
54+9a+3b-14-52=0
9a+3b-12=0
3a+b-4=0 .... eq 2

If you​ will subtract eq2 - eq1 You will get value of a that will be a-5
when u will have zero...
a-5=0
a=5

now you have to take a=5
2(5)+b(1)=0 
10+b=0
b= -10

Answer 2

Let f(x) = 2x^3 + ax^2 + bx - 14.

x - 2 = 0

x = 2.

When f(x) is divided by (x - 2), remainder = f(2).

f(2) = 2(2)^3 + a(2)^2 + b(2) - 14 = 0

       16 + 4a + 2b - 14 = 0

       4a + 2b + 2 = 0

        2a + b + 1 = 0

        2a + b = -1.   -------------- (1)


Now,

Given that when the expression is divided by (x-3), it leaves a remainder 52.

x - 3 = 0

x = 3.

f(3) = 2(3)^3 + a(3)^2 + b(3) - 14 = 52

       = 2 * 27 + 9a + 3b - 14 = 52

       = 54 + 9a + 3b - 14 = 52

      = 9a + 3b + 40 = 52

      = 9a + 3b = 52 - 40

      = 3a + b = 4   --------------- (2)


On solving (1) & (2), we get

2a + b = -1

3a + b = 4

------------------

-a = -5

a = 5

Substitute b = 5 in (1), we get

2a + b = -1

2(5) + b = -1

10 + b = -1

b = - 1 - 11

b = -11.


Therefore the values of a = 5 and b = -11.


Hope this helps