Solve x^6-35x^4+259x^2-225

$$x ^ { 6 } - 35 x ^ { 4 } + 259 x ^ { 2 } - 225$$

Factor

$$\left(x-5\right)\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+5\right)$$

Solution Steps

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

$$x^{6}-35x^{4}+259x^{2}-225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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^{6}-35x^{4}+259x^{2}-225 by x-1 to get x^{5}+x^{4}-34x^{3}-34x^{2}+225x+225. To factor the result, solve the equation where it equals to 0.

$$x^{5}+x^{4}-34x^{3}-34x^{2}+225x+225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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^{5}+x^{4}-34x^{3}-34x^{2}+225x+225 by x+1 to get x^{4}-34x^{2}+225. To factor the result, solve the equation where it equals to 0.

$$x^{4}-34x^{2}+225=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 225 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±225,±75,±45,±25,±15,±9,±5,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-34x^{2}+225 by x-3 to get x^{3}+3x^{2}-25x-75. To factor the result, solve the equation where it equals to 0.

$$x^{3}+3x^{2}-25x-75=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -75 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±75,±25,±15,±5,±3,±1$$

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.

$$x=-3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+3x^{2}-25x-75 by x+3 to get x^{2}-25. To factor the result, solve the equation where it equals to 0.

$$x^{2}-25=0$$

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, 0 for b, and -25 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-25\right)}}{2}$$

Do the calculations.

$$x=\frac{0±10}{2}$$

Solve the equation x^{2}-25=0 when ± is plus and when ± is minus.

$$x=-5 x=5$$

Rewrite the factored expression using the obtained roots.

$$\left(x-5\right)\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+5\right)$$

Evaluate

$$x^{6}-35x^{4}+259x^{2}-225$$