Chapter 4.6, Problem 11OE

To determine

To Find: Factor of polynomial $x^2-2x+1$

Answer to Problem 11OE

Factor of polynomial $x^2-2x+1=\left(x-1\right)\left(x-1\right)$

Explanation of Solution

Given information: polynomial $x^2-2x+1$

Formula used: If the quadratic trinomial $a{x}^2+bx+c$ can be factored into the product $\left(px+q\right)\left(rx+s\right)$ where $p,q,r,s$ are integers, then

  $a{x}^2+bx+c=pr{x}^2+\left(ps+qr\right)x+qs$

Setting corresponding coefficients equal gives

  $a=pr,b=ps+qr,c=qs$

Calculation:

polynomial $x^2-2x+1$

Since the coefficient of $x^2$ is $1$

So, $pr=1$

Therefore, let both pand r equal to $1$

So, that the factorization begins

  $\left(x\right)\left(x\right)$

Since the constant term is $1$

So, $qs=1$

There are two possibilities

  $\left(1\right)\left(1\right),\left(-1\right)\left(-1\right)$

Since the coefficient of $x$ is $-2$

So, $ps+qr=-2$ , but p and r are both $1$

So, $\begin{array}{l}ps+qr=-2\\ s+q=-2\end{array}$

Therefore, Choose the pair of factors having sum $-2:-1\text{ and }-1$

  $\therefore x^2-2x+1=\left(x-1\right)\left(x-1\right)$