Solve x^4-25x^2+4x^3-100x+3x^2-75

$$x ^ { 4 } - 25 x ^ { 2 } + 4 x ^ { 3 } - 100 x + 3 x ^ { 2 } - 75$$

Factor

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

Solution Steps

Multiply and combine like terms.

$$x^{4}+4x^{3}-22x^{2}-100x-75$$

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

$$x^{4}+4x^{3}-22x^{2}-100x-75=0$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+4x^{3}-22x^{2}-100x-75 by x+1 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.

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

Evaluate

$$\left(x+1\right)\left(x+3\right)\left(x^{2}-25\right)$$

Solution Steps

Combine -25x^{2} and 3x^{2} to get -22x^{2}.

$$x^{4}-22x^{2}+4x^{3}-100x-75$$