Question

Find the factorization of the polynomial below. 121x^2-22x+1

Answer 1

A little bit of trained eyeballing can help here. Let me tell you how I saw the answer without any computation.

Remember that the expansion of a squared binomial is

$(a+b)^2 = a^2+2ab+b^2$

So, the square of a binomial has the following properties, which you can easily spot:

- It has three terms

- Two of them are perfect squares

- The third is twice the product of the two roots.

Well, in this case we do have three terms, and indeed $121x^2$ is the square of 11x, and 1 is the square of itself. The only thing we have to check is that 22x is twice the product of 11x and 1, which is true.

So, the answer is

$121x^2-22x+1 = (11x-1)^2$

If you want to see some explicit calculations, just use the quadratic formula

$x_{1,2} = \cfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ to see that there is a double solution $x = \frac{1}{11}$, and then use the formula

$ax^2+bx+c = a(x-x_1)(x-x_2)$ to come to the same conclusion.

Answer 2

Hope this helps you!!..