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

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.