Solve 10x^4-3x^3-41x^2+12x+4

$$10 x ^ { 4 } - 3 x ^ { 3 } - 41 x ^ { 2 } + 12 x + 4$$

Factor

$$\left(x-2\right)\left(2x-1\right)\left(x+2\right)\left(5x+1\right)$$

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$10x^{4}-3x^{3}-41x^{2}+12x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 10. List all candidates \frac{p}{q}.

$$±\frac{2}{5},±\frac{4}{5},±2,±4,±\frac{1}{5},±1,±\frac{1}{10},±\frac{1}{2}$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 10x^{4}-3x^{3}-41x^{2}+12x+4 by x-2 to get 10x^{3}+17x^{2}-7x-2. To factor the result, solve the equation where it equals to 0.

$$10x^{3}+17x^{2}-7x-2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 10. List all candidates \frac{p}{q}.

$$±\frac{1}{5},±\frac{2}{5},±1,±2,±\frac{1}{10},±\frac{1}{2}$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 10x^{3}+17x^{2}-7x-2 by x+2 to get 10x^{2}-3x-1. To factor the result, solve the equation where it equals to 0.

$$10x^{2}-3x-1=0$$

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 10 for a, -3 for b, and -1 for c in the quadratic formula.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 10\left(-1\right)}}{2\times 10}$$

Do the calculations.

$$x=\frac{3±7}{20}$$

Solve the equation 10x^{2}-3x-1=0 when ± is plus and when ± is minus.

$$x=-\frac{1}{5} x=\frac{1}{2}$$

Rewrite the factored expression using the obtained roots.

$$\left(x-2\right)\left(2x-1\right)\left(x+2\right)\left(5x+1\right)$$

Evaluate

$$\left(2x-1\right)\left(5x+1\right)\left(x^{2}-4\right)$$