Question

If (x-2) is a factor of the expression 2x²+ax²+bx-14 and when the expression is divided by (x-3) it leaves a remainder 52,find the value of a and b.

Answer 1

$\huge{\mathfrak{\underline{Solution:}}}$

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.

Answer 2

Hi mate , here's ur solution,

Since (x-2) is a factor of 2x³+ax²+bx-14

=> 2(2)³ + a(2)² + b(2)-14=0

=>16+4a+2b-14=0

=>4a+2b=-2

=>2a+b=-1

Also when 2x³+ax²+bx-14 is divided by by x-3,it leaves remainder 52.

=> 2(3)³+a(3)²+b(3)-14=52

=>54+9a+3b=52+14

=>9a+3b=12

=>3a+b=4

Subtract i) from ii) we have,

a=5.

From i) we have ,

=>2(5)+b=-1

=>b=-11

Hence, the required value of a and b are 5 and -11 respectively!!!!!

Thanks!