Explain this---**Question Stem:** If p(x) = x^2 - 2√2x + 1, then p(√2) is equal to **Options:** A. 0 B. -1 C. 4√2 D. 8√2 + 1

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We have a polynomial function p of x equals x squared minus 2 root 2 x plus 1. We need to find p of root 2, which means substituting root 2 for every x in the polynomial. This gives us p of root 2 equals root 2 squared minus 2 root 2 times root 2 plus 1. Now let's evaluate each term systematically. The first term is root 2 squared, which equals 2. The second term is negative 2 root 2 times root 2, which equals negative 2 times 2, giving us negative 4. The third term is simply 1, which remains unchanged. Now we combine all the terms. We have p of root 2 equals 2 plus negative 4 plus 1. This simplifies to 2 minus 4 plus 1, which equals negative 1. Looking at the given options A through D, we can see that our answer negative 1 corresponds to option B. Therefore, the correct answer is B. Let's try an alternative approach using factoring. We might wonder if this polynomial is a perfect square trinomial. For a perfect square of the form a minus b squared, we have a squared minus 2ab plus b squared. If we try x minus root 2 squared, we get x squared minus 2 root 2 x plus 2. But this doesn't match our original polynomial, which has plus 1 at the end, not plus 2. So factoring doesn't work here, and our direct substitution method giving us negative 1 is indeed correct. Let's summarize our solution. We used direct substitution to evaluate p of root 2. We substituted root 2 into each term: root 2 squared equals 2, negative 2 root 2 times root 2 equals negative 4, and the constant term is 1. Combining these gives us 2 minus 4 plus 1, which equals negative 1. Therefore, the answer is option B. Key points to remember: substitute carefully, evaluate radicals correctly, and always check your arithmetic.