Solve f(x)=4x^3-24x^2+35x-12

$$f ( x ) = 4 x ^ { 3 } - 24 x ^ { 2 } + 35 x - 12$$

Factor

$$2\left(2x-3\right)\left(2x-1\right)\left(\frac{x}{2}-2\right)$$

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 -12 and q divides the leading coefficient 4. One such root is \frac{3}{2}. Factor the polynomial by dividing it by 2x-3.

$$\left(2x-3\right)\left(2x^{2}-9x+4\right)$$

Consider 2x^{2}-9x+4. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-9 ab=2\times 4=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -9.

$$a=-8 b=-1$$

Rewrite 2x^{2}-9x+4 as \left(2x^{2}-8x\right)+\left(-x+4\right).

$$\left(2x^{2}-8x\right)+\left(-x+4\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(x-4\right)-\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(2x-1\right)$$

Rewrite the complete factored expression.

$$\left(x-4\right)\left(2x-3\right)\left(2x-1\right)$$

Evaluate

$$4x^{3}-24x^{2}+35x-12$$