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  • Factorization of Perfect Squares

Expressions such as (x + y)2 and (x – y)2 are examples of perfect squares.

If we expand (x + y)2, we obtain (x + y)2 = (x + y)(x + y)

= x2 + xy + xy + y2

= x2 + 2xy + y2

The result of this equation may be stated as follows:

(1st term)2 + twice the product of the first term and the 2nd term + (2nd term)2

Similarly, if we expand (x – y)2, we obtain

(x – y)2 = (x – y)(x + y)

=x2 – xy – xy + y2

=x2 – 2xy + y2

The result is also:

(1st term)2 – twice the product of the first term and the 2nd term + (2nd term)2

It is important to know that expressions of the form x2 + 2xy + y2 and x2 – 2xy + y2 are also known as perfect squares.

Example 6.4.1:

Factorise the following:

(a) \( \scriptsize 16 \: – \: 8x \: + \: x^2 \)
(b) \( \scriptsize 4x^2 \: + \: 36x \: + \: 81 \)
(c) \( \scriptsize 1 \: – \: \normalsize \frac {2x}{3} \: + \: \frac {x^2}{9} \)
(d) \( \scriptsize 16x^2 \: – \: 64xy \: + \: 64y^2 \)

Solution

(a) \( \scriptsize 16 \: – \: 8x \: + \: x^2 \)

Rewrite as;

\( \scriptsize x^2 \: – \: 8x \: + \: 16 \)

 

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