Determine whether each statement is true or false. If false, explain why. Because x-1 is a factor of ƒ(x)=x^6-x^4+2x^2-2, we can also conclude that ƒ(1)=0

Verified step by step guidance

1. First, understand that if a polynomial function f(x) has a factor (x - a), then by the Factor Theorem, f(a) must equal 0.
2. Given that (x - 1) is a factor of the polynomial f(x) = x^6 - x^4 + 2x^2 - 2, apply the Factor Theorem by substituting x = 1 into the polynomial.
3. Substitute 1 into the polynomial: f(1) = 1^6 - 1^4 + 2(1^2) - 2.
4. Simplify the expression obtained in the previous step to check if it results in 0.
5. If f(1) equals 0, then the statement is true. If f(1) does not equal 0, then the statement is false and the explanation lies in the incorrect assumption or calculation.

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This means that if (x - c) is a factor of f(x), then substituting c into the polynomial will yield zero. In this case, if x - 1 is a factor of f(x), then f(1) must equal zero.

Polynomial Evaluation

Evaluating a polynomial involves substituting a specific value for the variable and calculating the result. For example, to evaluate f(x) = x^6 - x^4 + 2x^2 - 2 at x = 1, you would replace x with 1 and simplify the expression. This process is crucial for verifying claims about factors and roots of the polynomial.

Roots of a Polynomial

A root of a polynomial is a value of x that makes the polynomial equal to zero. Identifying roots is essential for understanding the behavior of the polynomial and its factors. If a polynomial has a factor (x - c), then c is a root, and this relationship is fundamental in polynomial factorization and analysis.