In Exercises 35–36, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x) = x^4 - 6x^3 + 14x^2 -14x + 5

Verified step by step guidance

1. Identify the constant term and the leading coefficient of the polynomial. Here, the constant term is 5 and the leading coefficient is 1.
2. List all factors of the constant term (5). The factors are ±1, ±5.
3. List all factors of the leading coefficient (1). The factors are ±1.
4. Use the Rational Zero Theorem, which states that any rational zero of the polynomial is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
5. Combine the factors to list all possible rational zeros: ±1, ±5.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem states that any rational solution (or zero) of a polynomial equation can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem helps in identifying all possible rational zeros of a polynomial function, which can then be tested to find actual zeros.

Factors of a Polynomial

Factors of a polynomial are the numbers or expressions that can be multiplied together to yield the polynomial. For the Rational Zero Theorem, identifying the factors of the constant term (the last term of the polynomial) and the leading coefficient (the coefficient of the highest degree term) is essential, as these factors determine the potential rational zeros.

Testing for Zeros

Once the possible rational zeros are identified using the Rational Zero Theorem, each candidate must be tested in the polynomial function to determine if it is indeed a zero. This is typically done by substituting the candidate into the polynomial and checking if the result equals zero, which confirms that the candidate is a valid zero of the function.