Question

If f(x)(2x^3+x^2+bx-6) leaves the remainder 36 ,when divided by (x-3),find the value of b.With the value of b,factorise f (x)

Answer 1

Given,
f(x) = 2x³ + x²+bx-6
According to question,
if f(x) i.e 2x³ + x²+bx-6 is divided by x-3 then it leaves a reminder of 36.
so ,
2x³ + x² + bx - 6 - 36 will be divisible by x-3.
so that
(x-3) is an factor of 2x³ + x²+bx-42 = 0. ---------------------(i)
∴ x - 3 =0
   x = 3
put the value of x in eqn(i).
    2(3)³ + 3²+3b-42 = 0
⇒ 2*27 + 9 + 3b -42 =0
⇒ 54 + 9 -42 + 3b = 0
⇒ 21 = -3b
∴ b = - 21/3 = -7.

Now,
f(x) = 2x³ + x² - 7x - 6  -----------------------------(ii)
put x = 2;
f(x) = 2(2)³ + 2² - 7*2 - 6 = 16+4-14-6 = 0
so the one factor of the f(x) is x-2.

Divide the eqn(ii) by (x-2) to find out remaining two factors.
           
                                x-2) 2x³ + x² - 7x - 6 (2x² + 5x +3
                                     -(2x³ - 4x²)
                                ------------------------
                                          0  + 5x² - 7x - 6
                                              - (5x² - 10x)
                                         ----------------------
                                                    0 + 3x - 6
                                                       - (3x - 6)
                                               ------------------
                                                         0 + 0
                                 
Hence another factor will be 2x² + 5x +3 but, it's in a quadratic form so solve this quadratic equation to get the factors.
    2x² + 5x + 3 = 0
⇒ 2x² + 3x + 2x +3 = 0
⇒ x(2x +3) +1(2x + 3) = 0
⇒ (x + 1)(2x + 3) = 0
                                      |
                 x + 1 = 0       |    2x + 3  = 0
                   x  =  - 1       |          2x = -3
                                      |             x = -3/2
Therefore the factors are 2, -1 & -3/2
And, Value of b = -7

Answer 2

Answer:k=-7

Step-by-step explanation: