Factor polynomial $$ -4x^{4}+x^{2}-2x+1 $$

$$ -4x^{4}+x^{2}-2x+1 = -( x+1 )( 2x-1 )( 2x^{2}-x+1 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{4}+x^{2}-2x+1 = - ~ ( 4x^{4}-x^{2}+2x-1 ) $$

Step 2 :

The polynomial $ 4x^{4}-x^{2}+2x-1 $ can be factorized by using the rational root test. To apply this test first we need to find at least one rational zero.
Polynomial roots calculator can be used to find rational zero.

In this case one rational zero is $ x = -1 $.

This means that we can divide polynomial $ 4x^{4}-x^{2}+2x-1 $ by $ x+1 $ (the factor theorem)

After division we have: (you can use Synthetic Division Calculator for the step-by-step explanation on how to divide polynomials. )

$$ \frac{ 4x^{4}-x^{2}+2x-1 }{ x+1 } = 4x^{3}-4x^{2}+3x-1 $$

that is:

$$ 4x^{4}-x^{2}+2x-1 = ( x+1 )( 4x^{3}-4x^{2}+3x-1 ) $$

Step 3 :

Use rational root test to get

$$ 4x^{3}-4x^{2}+3x-1 = ( 2x-1 )( 2x^{2}-x+1 ) $$

Step 4 :

Step 4: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 2 , b = -1 ~ \text{ and } ~ c = 1 $$

Step 5: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 2 $$

Step 6: Find out two numbers that multiply to $ a \cdot c = 2 $ and add to $ b = -1 $.

Step 7: All pairs of numbers with a product of $ 2 $ are:

PRODUCT = 2
1     2	-1     -2

Step 8: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

Step 9: Because none of these pairs will give us a sum of $ \color{blue}{ -1 }$, we conclude the polynomial cannot be factored.

Polynomial Factoring Calculator