Which option is the simplified form of (x^8 - 81) / (x^4 - 9), where x^4 ≠ 9?
Answer and explanation
Correct answer: x^4 + 9
The governing concept is the difference-of-squares identity, u^2 - v^2 = (u-v)(u+v). Take u = x^4 and v = 9. Then x^8 - 81 = (x^4)^2 - 9^2 = (x^4 - 9)(x^4 + 9). Substituting this factorisation into the fraction gives [(x^4 - 9)(x^4 + 9)]/(x^4 - 9). Because x^4 ≠ 9, the denominator is non-zero and cancellation is valid, leaving x^4 + 9. Therefore option B is correct. Option A is the cancelled factor, option C uses the wrong power, and option D does not result from the difference-of-squares identity.
Frequently asked questions
What is the correct answer to this question?
x^4 + 9
Why is this the correct answer?
The governing concept is the difference-of-squares identity, u^2 - v^2 = (u-v)(u+v). Take u = x^4 and v = 9. Then x^8 - 81 = (x^4)^2 - 9^2 = (x^4 - 9)(x^4 + 9). Substituting this factorisation into the fraction gives [(x^4 - 9)(x^4 + 9)]/(x^4 - 9). Because x^4 ≠ 9, the denominator is non-zero and cancellation is valid, leaving x^4 + 9. Therefore option B is correct. Option A is the cancelled factor, option C uses the wrong power, and option D does not result from the difference-of-squares identity.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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