Solve f(x)=x^4-3x^3-3x^2-75x-700

$$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 3 x ^ { 2 } - 75 x - 700$$

Factor

$$\left(x-7\right)\left(x+4\right)\left(x^{2}+25\right)$$

Solution Steps

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

$$x^{4}-3x^{3}-3x^{2}-75x-700=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -700 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±700,±350,±175,±140,±100,±70,±50,±35,±28,±25,±20,±14,±10,±7,±5,±4,±2,±1$$

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.

$$x=-4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-3x^{3}-3x^{2}-75x-700 by x+4 to get x^{3}-7x^{2}+25x-175. To factor the result, solve the equation where it equals to 0.

$$x^{3}-7x^{2}+25x-175=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -175 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±175,±35,±25,±7,±5,±1$$

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.

$$x=7$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-7x^{2}+25x-175 by x-7 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.

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

Do the calculations.

$$x=\frac{0±\sqrt{-100}}{2}$$

Polynomial x^{2}+25 is not factored since it does not have any rational roots.

$$x^{2}+25$$

Rewrite the factored expression using the obtained roots.

$$\left(x-7\right)\left(x+4\right)\left(x^{2}+25\right)$$

Evaluate

$$\left(x-7\right)\left(x+4\right)\left(x^{2}+25\right)$$