If p(x) = x² − 2√2x + 2, which is the correct factorized form of p(x)?
Answer and explanation
Correct answer: (x − √2)²
Use the perfect-square identity (x − a)² = x² − 2ax + a². Taking a = √2 gives (x − √2)² = x² − 2√2x + (√2)² = x² − 2√2x + 2, which is exactly the given polynomial. Therefore option A is correct. Option B expands to x² + 2√2x + 2, so its middle-term sign is wrong. Option C expands to x² − 4x + 4 and has different coefficients. Option D uses the difference-of-squares identity and gives x² − 2, which lacks the required middle term. The factorization also shows that √2 is a repeated zero, because the same linear factor occurs twice.
Frequently asked questions
What is the correct answer to this question?
(x − √2)²
Why is this the correct answer?
Use the perfect-square identity (x − a)² = x² − 2ax + a². Taking a = √2 gives (x − √2)² = x² − 2√2x + (√2)² = x² − 2√2x + 2, which is exactly the given polynomial. Therefore option A is correct. Option B expands to x² + 2√2x + 2, so its middle-term sign is wrong. Option C expands to x² − 4x + 4 and has different coefficients. Option D uses the difference-of-squares identity and gives x² − 2, which lacks the required middle term. The factorization also shows that √2 is a repeated zero, because the same linear factor occurs twice.
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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