Question

(x-2) is a factor of 2x^3+ax^2+bx-2 and when this expression is divided by (x+3), the remainder is -50. Find the value a and b and the other factors. Hence solve the equation 2x^3+ax^2+bx=2

Answer 1

a = -1, b = -5

Solution to the equation :

x = {-1, -0.5, 2}

Explanation:

By the factor theorem, if x-2 is a factor of f(x) the f(2) = 0, so

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

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

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

---> 2a + b = -7 (A)

Also by the remainder theorem:

2(-3)^3 + (-3)^2a - 3b - 2 = -50

---> -54 + 9a -3b - 2 = -50

---> 9a - 3b = 6

---> 3a - b = 2 (B)

Solving the 2 equation A and B:

Adding A and B gives:

5a = -5

--> a = -1.

Substituting for a in equation A:

2(-1) + b = -7

--> b = -5.

2x^3+ax^2+bx=2

--> 2x^3 - 1x^2 - 5x - 2 = 0

One solution to this is x = 2 because x-2 is a factor.

Finding the other solutions:

Dividing by x-2:

x - 2) 2x^3 - 1x^2 - 5x - 2(2x^2 + 3x + 1 <------- Quotient

2x^3 - 4x^2

3x^2 -5x

3x^2 - 6x

x - 2

x - 2

.......

So: 2x^2 + 3x + 1 = (2x + 1)(x + 1)

Thus:

2x^3 - 1x^2 - 5x - 2 = (x - 2)(2x + 1)(x + 1)

So the solution to the original equation is

x = -1, -0.5, 2.

Answer 2

f(-3) = 2(-3)³ + a(-3)² + b(-3) - 2 = -50

2(-27) + 9a - 3b - 2 = -50

-54 + 9a - 3b - 2 = -50

9a - 3b = 6

3a - b = 2

f(x) = 2x³ + ax² + (3a - 2)x - 2

f(2) = 2(2³) + a(2²) + (3a - 2)(2) - 2 = 0

16 + 4a + 6a - 4 - 2 = 0

10a + 10 = 0

a = -1, b = -5

f(x) = 2x³ - x² - 5x - 2

= (x - 2)(2x² + 3x + 1)

= (x - 2)(2x + 1)(x + 1)

See the attachment.