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x^4-36 factor each polynomial completely. make sure to chek for a gcf first
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GPT-5 mini
AI
Step 1 — GCF: none (GCF = 1).
Step 2 — difference of squares:
x^4 - 36 = (x^2)^2 - 6^2 = (x^2 - 6)(x^2 + 6).
Step 3 — factor further if allowed:
- Over the rationals (or integers): (x^2 - 6)(x^2 + 6) — stops here.
- Over the real numbers: x^2 - 6 factors as (x - √6)(x + √6), so
x^4 - 36 = (x - √6)(x + √6)(x^2 + 6).
(Over the complex numbers you can factor x^2 + 6 = (x - i√6)(x + i√6) as well.)
Step 2 — difference of squares:
x^4 - 36 = (x^2)^2 - 6^2 = (x^2 - 6)(x^2 + 6).
Step 3 — factor further if allowed:
- Over the rationals (or integers): (x^2 - 6)(x^2 + 6) — stops here.
- Over the real numbers: x^2 - 6 factors as (x - √6)(x + √6), so
x^4 - 36 = (x - √6)(x + √6)(x^2 + 6).
(Over the complex numbers you can factor x^2 + 6 = (x - i√6)(x + i√6) as well.)
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