Solve f(x)=16x^3-24x^2+9x-1

$$f ( x ) = 16 x ^ { 3 } - 24 x ^ { 2 } + 9 x - 1$$

Factor

$$\left(x-1\right)\left(4x-1\right)^{2}$$

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 -1 and q divides the leading coefficient 16. One such root is \frac{1}{4}. Factor the polynomial by dividing it by 4x-1.

$$\left(4x-1\right)\left(4x^{2}-5x+1\right)$$

Consider 4x^{2}-5x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-5 ab=4\times 1=4$$

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 4.

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

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite 4x^{2}-5x+1 as \left(4x^{2}-4x\right)+\left(-x+1\right).

$$\left(4x^{2}-4x\right)+\left(-x+1\right)$$

Factor out 4x in the first and -1 in the second group.

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

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

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

Rewrite the complete factored expression.

$$\left(x-1\right)\left(4x-1\right)^{2}$$

Evaluate

$$\left(x-1\right)\left(4x-1\right)^{2}$$