Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x⁶-x⁴+2x²-2, we can also conclude that ƒ(1) = 0.

Verified step by step guidance

1. Recall the Factor Theorem, which states that if \( x - a \) is a factor of a polynomial \( f(x) \), then \( f(a) = 0 \).
2. Identify the value of \( a \) from the factor \( x - 1 \). Here, \( a = 1 \).
3. Evaluate the polynomial \( f(x) = x^6 - x^4 + 2x^2 - 2 \) at \( x = 1 \) by substituting 1 into the polynomial: \( f(1) = 1^6 - 1^4 + 2(1)^2 - 2 \).
4. Simplify the expression \( f(1) \) step-by-step to check if it equals zero.
5. If \( f(1) = 0 \), then \( x - 1 \) is a factor and the statement is true; if not, the statement is false because the Factor Theorem requires \( f(1) = 0 \) for \( x - 1 \) to be a factor.

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has (x - c) as a factor if and only if f(c) = 0. This means that if x - 1 is a factor of f(x), then substituting x = 1 into f(x) should yield zero.

Evaluating Polynomials

Evaluating a polynomial at a specific value involves substituting that value for the variable and simplifying. This process helps determine if the polynomial equals zero at that point, which is essential for applying the Factor Theorem.

True/False Reasoning in Algebra

Determining the truth of a statement in algebra requires verifying conditions precisely. If a statement claims a factor implies a certain value of the polynomial, checking the polynomial's value at that point confirms or refutes the claim.