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The First and Second Algebraic Identity


You may have previously seen that (a × b)2 = a2 × b2, but what is (a+b)2? If you write (a + b)2 as (a + b)(a + b), you can use what you learned about the distributive property of multiplying parentheses to find the answer: (a + b)2 = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2

This also applies to (a b)2: (a b)2 = (a b)(a b) = a2 ab ba + b2 = a2 2ab + b2

These results are used all the time in mathematics, and we call them the first and second algebraic identities. Learn these, and you’ll save a lot of time you would’ve spent calculating!

Rule

The First Algebraic Identity

(a + b)2 = a2 + 2ab + b2

The Second Algebraic Identity

(a b)2 = a2 2ab + b2

Example 1

Expand (x 2)2 + (3 + x)2

If you apply the first and second algebraic identities to the squared parentheses, you get

= (x 2)2 + (3 + x)2 = (x2 4x + 4) + (9 + 6x + x2) = 2x2 + 2x + 13

(x 2)2 + (3 + x)2 = (x2 4x + 4) + (9 + 6x + x2) = 2x2 + 2x + 13

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The Third Algebraic Identity