Solve 6x^3+35x^2-52x-21=0

$$6 x ^ { 3 } + 35 x ^ { 2 } - 52 x - 21 = 0$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x=-7$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -21 and q divides the leading coefficient 6. List all candidates \frac{p}{q}.

$$±\frac{7}{2},±7,±\frac{21}{2},±21,±\frac{7}{6},±\frac{7}{3},±\frac{1}{2},±1,±\frac{3}{2},±3,±\frac{1}{6},±\frac{1}{3}$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-7$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 6x^{3}+35x^{2}-52x-21 by x+7 to get 6x^{2}-7x-3. Solve the equation where the result equals to 0.

$$6x^{2}-7x-3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 6 for a, -7 for b, and -3 for c in the quadratic formula.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\left(-3\right)}}{2\times 6}$$

Do the calculations.

$$x=\frac{7±11}{12}$$

Solve the equation 6x^{2}-7x-3=0 when ± is plus and when ± is minus.

$$x=-\frac{1}{3} x=\frac{3}{2}$$

List all found solutions.

$$x=-7 x=-\frac{1}{3} x=\frac{3}{2}$$