In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x^4−2x^3+x^2+12x+8

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = x^4 - 2x^3 + x^2 + 12x + 8 \).
2. Apply the Rational Zero Theorem to list all possible rational zeros. These are the factors of the constant term (8) divided by the factors of the leading coefficient (1).
3. Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros. Count the sign changes in \( f(x) \) for positive zeros and in \( f(-x) \) for negative zeros.
4. Use a graphing utility to plot the polynomial function and visually identify any real zeros. This can help in selecting a candidate for synthetic division.
5. Perform synthetic division using one of the potential zeros identified from the graph or Rational Zero Theorem. If the remainder is zero, the candidate is a root, and the quotient is a reduced polynomial. Repeat the process on the reduced polynomial to find all zeros.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem provides a method for identifying possible rational roots of a polynomial function. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps narrow down the candidates for testing potential zeros of the polynomial.

Descartes's Rule of Signs

Descartes's Rule of Signs is a technique used to determine the number of positive and negative real roots of a polynomial. By counting the number of sign changes in the polynomial's coefficients, one can estimate how many positive roots exist. Similarly, by substituting -x for x and counting sign changes, one can find the number of negative roots, aiding in the overall analysis of the polynomial's roots.

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions. By plotting the polynomial function, one can observe where the graph intersects the x-axis, indicating the real roots of the polynomial. This visual representation can provide insights into the behavior of the function and assist in confirming the roots found through algebraic methods.