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Some products show up so often they get shortcuts. Squaring a binomial and multiplying conjugates each follow a fixed pattern you can write down without full FOIL.
Recognizing these patterns saves time and prevents the classic mistake of thinking equals — it does not.
The squared binomial
— the square of the first, plus twice the product, plus the square of the last. The middle term is the one people forget.
works the same with a minus on the middle term. Never drop the .
Difference of squares
. The two middle terms cancel, leaving only the squares.
Spotting a sum times a difference of the same two terms lets you write the answer instantly — no FOIL needed.
Worked examples
Example 1: squaring a binomial
Multiply .
Answer:
Example 2: difference of squares
Multiply .
Answer:
Example 3: squaring a difference
Multiply .
Answer:
Try one yourself
Common questions
Why isn't equal to ?
Squaring a binomial by FOIL produces a middle term: . The cannot be dropped.
When does difference of squares apply?
When you multiply a sum and a difference of the same two terms, like . The result is .
Are these just FOIL shortcuts?
Yes. You can always FOIL, but recognizing the pattern lets you write the answer immediately and avoid sign errors.
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