find an interval of unit length which contains the negative root of smallest magnitude of the equation $3x^3+10x^2+10x+7=0$

Question

I don't know how am I supposed to do it? Kindly help me with it. I have tried using bisection method but I don't have an idea regarding this.

For a cubic, the derivative is quadratic, which is easy to solve. By Rolle's theorem, between two roots of the cubic, there must be a root of its derivative. Moreover, if there are two real roots of the cubic, then all roots are real. This implies that if you evaluate the cubic at the roots of its derivative you get two values that multiplied are $\leq 0$, when there are more than one real root of the cubic. Therefore, the values of the cubic at the roots of its derivative tell you both the existence of more than one root and serve to bracket the roots. — user647486, Apr 12 '19 at 17:48

score 1 · Answer 1 · answered Apr 12 '19 at 17:43

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Note that $$3x^3+10x^2+10x+7=(7+3x)(x^2+x+1)$$

answered Apr 12 '19 at 17:43

1 Answers1