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Flip to reveal answersDifference between = and ≡?
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All 10 Flashcards — Proving identities
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Question
Difference between = and ≡?
Answer
= (equation) is true for particular values you solve for; ≡ (identity) is true for ALL values, which you prove.
Question
How do you prove an identity LHS ≡ RHS?
Answer
Transform ONE side (the busier one) step by step into the other. Never move terms across the ≡.
Question
Why can't you prove an identity by substituting one value?
Answer
One value only checks that single case; an identity must hold for every x.
Question
Which side should you start from?
Answer
The busier side — the one with brackets/powers to expand or fractions to combine.
Question
Method for a polynomial identity?
Answer
Expand all brackets, then collect like terms until it matches the target.
Question
What does (a − b)² expand to?
Answer
a² − 2ab + b² — don't forget the middle term −2ab.
Question
Method for a rational (fraction) identity?
Answer
Put the side over a common denominator, combine the numerators, then simplify/factor.
Question
Common denominator of 1/x and 1/(x + 1)?
Answer
x(x + 1) — the product of the two distinct denominators.
Question
What does "hence" tell you in a later part?
Answer
Use the identity/result you just proved — don't re-derive it from scratch.
Question
Prove (x + 3)² − (x − 3)² ≡ 12x.
Answer
Expand: (x² + 6x + 9) − (x² − 6x + 9) = 12x. The x² and 9 cancel.
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