Solve f(x)=7x^4-58x^3+60x^2+250x+125

$$f ( x ) = 7 x ^ { 4 } - 58 x ^ { 3 } + 60 x ^ { 2 } + 250 x + 125$$

Factor

$$\left(x+1\right)\left(7x+5\right)\left(x-5\right)^{2}$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 125 and q divides the leading coefficient 7. One such root is -\frac{5}{7}. Factor the polynomial by dividing it by 7x+5.

$$\left(7x+5\right)\left(x^{3}-9x^{2}+15x+25\right)$$

Consider x^{3}-9x^{2}+15x+25. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 25 and q divides the leading coefficient 1. One such root is 5. Factor the polynomial by dividing it by x-5.

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

Consider x^{2}-4x-5. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-5\right)=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite x^{2}-4x-5 as \left(x^{2}-5x\right)+\left(x-5\right).

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

Factor out x in x^{2}-5x.

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

Factor out common term x-5 by using distributive property.

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

Rewrite the complete factored expression.

$$\left(x+1\right)\left(7x+5\right)\left(x-5\right)^{2}$$

Evaluate

$$\left(x+1\right)\left(7x+5\right)\left(x-5\right)^{2}$$