If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 is evaluated at x = 1, what

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Question: If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 is evaluated at x = 1, what is the result?

Options:

Correct Answer: 1

Solution:

Substituting x = 1 into the polynomial gives f(1) = 1 - 4 + 6 - 4 + 1 = 0.

If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 is evaluated at x = 1, what is the result?

If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 is evaluated at x = 1, what is the result?

Step 1: Write down the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1.
Step 2: Substitute x = 1 into the polynomial. This means replacing every x in the polynomial with 1.
Step 3: Calculate each term: 1^4 = 1, -4 * 1^3 = -4, 6 * 1^2 = 6, -4 * 1 = -4, and 1 = 1.
Step 4: Now, add all the results together: 1 - 4 + 6 - 4 + 1.
Step 5: Perform the addition step-by-step: 1 - 4 = -3, then -3 + 6 = 3, then 3 - 4 = -1, and finally -1 + 1 = 0.
Step 6: The final result is 0.

Polynomial Evaluation – The process of substituting a specific value into a polynomial to find its output.
Understanding Polynomial Structure – Recognizing the coefficients and terms of a polynomial and how they contribute to the overall value when evaluated.