Factorise each of the following:
$9 x^{2} - 6 x + 1$

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Solution

In the given polynomial $9 x^{2} - 6 x + 1$ ,
The first term is $9 x^{2}$ and its coefficient is $9$ .
The middle term is $- 6 x$ and its coefficient is $- 6$ .
The last term is a constant term $1$ .

Multiplication of the coefficient of the first term by the constant is 9$.

We now find the factor of $9$ whose sum equals the coefficient of the middle term, which is $- 6$ and then factorize the polynomial $9 x^{2} - 6 x + 1$ as shown below:
$9 x^{2} - 6 x + 1 = 9 x^{2} - 3 x - 3 x + 1 = 3 x (3 x - 1) - 1 (3 x - 1) = (3 x - 1) (3 x - 1) = (3 x - 1)^{2}$

Hence, $9 x^{2} - 6 x + 1 = (3 x - 1)^{2}$ .

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