Question

The polynomial (2x - 1)(x^2- 2) - x(x^2 - x - 2) can be written in the form ax^3 + bx^2 + cx + d. a, b, c, d are constants. What are the values of a, b, c, d?

Answer 1

Given the following expression:

`(2x-1)(x^2\text{ - 2) - x(}x^2\text{ - x - 2)}`

To be able to find the value of the constants a, b, c, and d, we must first simplify the expression in the form of ax^3 + bx^2 + cx + d.

We get,

`(2x-1)(x^2\text{ - 2) - x(}x^2\text{ - x - 2)}`
`\text{ (2x}^3-4x-x^2+2)-(x^3-x^2\text{ - 2)}`
`\text{ 2x}^3-4x-x^2+2-x^3+x^2+2=2x^3-x^3-x^2+x^2\text{ - 4x + 2 + 2}`
`\text{ = x}^3+0x^2\text{ - 4x + 4}`

Therefore, (2x - 1)(x^2- 2) - x(x^2 - x - 2) when simplified is x^3 + 0x^2 - 4x + 4.

Following the standard form ax^3 + bx^2 + cx + d, the following constants are:

ax^3 = x^3 = 1x^3

a = 1

bx^2 = 0x^2

b = 0

cx = - 4x

c = -4

d = 4

In Summary: a = 1, b = 0, c = -4 and d = 4