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How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

We can see that 104 = 2^3 * 13 = 2*2*26, 30 = 2 + 2 + 26, and 108 = 2*2 + 2*26 + 2*26, so the coefficients agree with the Vieta's formulas, so the roots of the equation above are 2, 2, 26. In conclusion, it has 2 distinct real roots. 

Alternatively, we can try to factorise the polynomial. This can be done by (x-2)^2*(x-26), and so we can see that the equation has 2 distinct real roots. 

Andreea I. Uni Admissions Test .MAT. tutor, Uni Admissions Test .STEP...

8 months ago

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