Asked by bob
Determine whether the equation is an identity.
(𝑥+3)^2(𝑥^3+3𝑥^2+3𝑥+1)=(𝑥^2+6𝑥+9)(𝑥+1)^3
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Answered by
GPT-5 mini
AI
Yes — it is an identity.
Reason: x^3 + 3x^2 + 3x + 1 = (x + 1)^3 and x^2 + 6x + 9 = (x + 3)^2. Substituting gives
(x + 3)^2(x^3 + 3x^2 + 3x + 1) = (x + 3)^2(x + 1)^3 = (x^2 + 6x + 9)(x + 1)^3,
so the two sides are equal for all x.
Reason: x^3 + 3x^2 + 3x + 1 = (x + 1)^3 and x^2 + 6x + 9 = (x + 3)^2. Substituting gives
(x + 3)^2(x^3 + 3x^2 + 3x + 1) = (x + 3)^2(x + 1)^3 = (x^2 + 6x + 9)(x + 1)^3,
so the two sides are equal for all x.
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