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State the possible rational roots of 2x^3-4x^2+8x-10=0

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2 Answers

     2x3 - 4x2 + 8x - 10 = 0

By the Rational Zeros Theorem, we can find all the possible rational zeros (or roots) of a polynomial function, such as this one, by the following:

     Possible rational zeros  =  (factors of the constant term) / (factors of the leading coefficient)

Make a list of all the possible factors of the constant term in the equation (which, in this case, is -10) and all the possible factors of the leading coefficient (which, in this case, is 2).

        Factors of constant term, -10:       ±1 , ±2 , ±5 , ±10 

     Factors of leading coefficient, 2:       ±1 , ±2

Possible rational zeros =  ±1, ±2, ±5, ±10 / ±1, ±2

                                 =  ±1/±1 , ±1/±2 , ±2/±1 , ±2/±2 , ±5/±1 , ±5/±2 , ±10/±1 , ±10/±2

                                 =  ±1 , ±1/2 , ±2 , ±1 , ±5 , ±5/2 , ±10 , ±5

                                 =  ±1 , ±2 , ±5 , ±10 , ±1/2 , ±5/2

Thus, you have the following 12 possible rational zeros/roots for this polynomial function:     

                           ± 1  ,  ± 2  ,  ± 5  ,  ± 10  ,  ± 1/2  ,  ± 5/2

Hello Connor,

The constant term of this polynomial is 10, with factors 1, 2, 5, and 10.

The leading coefficient is 2, with factors 1 and 2.

Then the Rational roots yields the following possible solutions:

±(factors of 10)/(factors of 2)

= ±(1, 2, 5, 10)/(1,2)

= ±1/1, ±1/2, ±2/1, ±5/1, ±5/2, ±10/1

±1, ±1/2, ±2, ±5, ±5/2, ±10 (answers)

Hope this helps.