Solve f(x)=12x^3-17x^2-20x+25

$$f ( x ) = 12 x ^ { 3 } - 17 x ^ { 2 } - 20 x + 25$$

Factor

$$\left(x-1\right)\left(3x-5\right)\left(4x+5\right)$$

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 25 and q divides the leading coefficient 12. One such root is \frac{5}{3}. Factor the polynomial by dividing it by 3x-5.

$$\left(3x-5\right)\left(4x^{2}+x-5\right)$$

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

$$a+b=1 ab=4\left(-5\right)=-20$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.

$$-1,20 -2,10 -4,5$$

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 1.

$$a=-4 b=5$$

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

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

Factor out 4x in the first and 5 in the second group.

$$4x\left(x-1\right)+5\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(4x+5\right)$$

Rewrite the complete factored expression.

$$\left(3x-5\right)\left(x-1\right)\left(4x+5\right)$$

Evaluate

$$\left(x-1\right)\left(3x-5\right)\left(4x+5\right)$$