List the potential rational zeros of the following function. Please explain. f(x)=-4x^4+3x^2-2x^2-2x+6 not calculus

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The rational root theorem says that any rational roots of a polynomial a_nx^n+a_(n-1)x^(n-1)+...+a_1x+a_0 = 0 are rational numbers x=+-p/q where p is a factor of a_0 and q is a factor of a_n

In this case a_0 = 6 and a_n = -4

Factors of 6 are 1, 2, 3, and 6

Facotrs of 4 are 1, 2, 4

So the possible rational roots are

`x = +- 1/1= +-1, +- 2/1 = +-2, +- 3/1 = +- 3, +- 6/1 = +- 6, `

`+- 1/2 = +- 1/2, +- 2/2 = +- 1, +- 3/2 = +- 3/2, +- 6/2 = +- 3, `

`+- 1/4 = +- 1/4, +- 2/4 = +- 1/2, +- 3/4 = +- 3/4, +- 6/4 = +- 3/2`

Getting rid of duplicates we get

`x = +-1, +-2, +- 3, +- 6, +- 1/2, +- 3/2, +- 1/4, +- 3/4`

 

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