Solve f(x)=x^4+3x^3-11x^2-3x+10

$$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 11 x ^ { 2 } - 3 x + 10$$

Factor

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

Solution Steps

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

$$x^{4}+3x^{3}-11x^{2}-3x+10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 10 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±10,±5,±2,±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+3x^{3}-11x^{2}-3x+10 by x-1 to get x^{3}+4x^{2}-7x-10. To factor the result, solve the equation where it equals to 0.

$$x^{3}+4x^{2}-7x-10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -10 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±10,±5,±2,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+4x^{2}-7x-10 by x+1 to get x^{2}+3x-10. To factor the result, solve the equation where it equals to 0.

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

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

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

Do the calculations.

$$x=\frac{-3±7}{2}$$

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

$$x=-5 x=2$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

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