Solve F(x)=-9x^4+39x^3-25x^2-43x+30

$$F ( x ) = - 9 x ^ { 4 } + 39 x ^ { 3 } - 25 x ^ { 2 } - 43 x + 30$$

Factor

$$\left(3-x\right)\left(3x-5\right)\left(3x-2\right)\left(x+1\right)$$

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 30 and q divides the leading coefficient -9. One such root is \frac{5}{3}. Factor the polynomial by dividing it by 3x-5.

$$\left(3x-5\right)\left(-3x^{3}+8x^{2}+5x-6\right)$$

Consider -3x^{3}+8x^{2}+5x-6. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient -3. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(-3x^{2}+11x-6\right)$$

Consider -3x^{2}+11x-6. Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=11 ab=-3\left(-6\right)=18$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=9 b=2$$

Rewrite -3x^{2}+11x-6 as \left(-3x^{2}+9x\right)+\left(2x-6\right).

$$\left(-3x^{2}+9x\right)+\left(2x-6\right)$$

Factor out 3x in the first and -2 in the second group.

$$3x\left(-x+3\right)-2\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(3x-2\right)$$

Rewrite the complete factored expression.

$$\left(3x-5\right)\left(3x-2\right)\left(x+1\right)\left(-x+3\right)$$

Evaluate

$$\left(3-x\right)\left(3x-5\right)\left(3x-2\right)\left(x+1\right)$$