Question

Consider the polynomial g(x) = 4x³ - x2 - 24x+6

a. List all possible rational zeros of the polynomial. (Assume all numbers shown are both + and -).
O1, 2, 4, 1/2, 1/3, 2/3, 4/3, 1/6
1, 2, 3, 4, 6, 1/2, 3/2, 1/3, 2/3, 4/3, 1/4, 3/4, 1/6
O 1, 2, 3, 4, 6, 1/2, 3/2, 1/4, 3/4
O1, 2, 3, 6, 1/2, 3/2, 1/4, 3/4
b. List all actual zeros of the polynomial (rational, irrational, real, and complex).
Give exact answers, separated by commas as necessary.
Zeros:

Answer 1

a. x = 1/4

b. x =
`+√(6)`,
`-√(6)`, 1/4

Explanation:

a. I factorised the polynomial.

`g(x) = 4x^3 - x^2 -24x +6 = 0`

`-x^2 (-4x + 1) +6 (-4x +1)`

`(-4x+1)(-x^2+6)`

Solved for the roots.

`-4x + 1 = 0`

`-4x = -1`

`x = (1)/(4)`

`-x^2+6 = 0`

`-x^2 = -6`

`x^2 = 6`

`x = +√(6)`

`x=-√(6)`

A rational number is a real number that's digits terminate and/or repeat. Therefore, 1/4 is the only rational zero.

b. All of the zeros we found in part A. are
`+√(6), -√(6), and (1)/(4)`