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A comprehensive guide to finding zeros of polynomial functions. It covers key concepts like the remainder theorem, factor theorem, rational zero theorem, and descartes' rule of signs. Illustrative examples and practice problems to reinforce understanding. It is suitable for students studying algebra or precalculus.
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The Remainder Theorem If a polynomial f (x) is divided by x − k, then the remainder is the value f (k). How To … Given a polynomial function f, evaluate f (x) at x = k using the Remainder Theorem.
EX. Use the Remainder Theorem to evaluate f (x) = 6x^4 – x^3 − 15x^2 + 2x − 7 at x = 2. Try It # Use the Remainder Theorem to evaluate f (x) = 2x 5 − 3x 4 − 9x 3
Try It # Use the Factor Theorem to find the zeros of f (x) = x^3 + 4x^2 − 4x − 16 given that (x − 2) is a factor of the polynomial
EX. List all possible rational zeros of f (x) = 2x 4 − 5x 3
Try It # Use the Rational Zero Theorem to find the rational zeros of f (x) = x^3 − 5x^2 + 2x + 1.
Finding the Zeros of Polynomial Functions The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function. How To … Given a polynomial function f, use synthetic division to find its zeros.
The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations. EX. Find the zeros of f (x) = 3x 3
Try It # Find the zeros of f (x) = 2x^3 + 5x^2 − 11x + 4.
Try It # Find a third degree polynomial with real coefficients that has zeros of 5 and −2i such that f (1) = 10.
If the polynomial is written in descending order, Descartes’ Rule of Signs tells us of a relationship between the number of sign changes in f (x) and the number of positive real zeros. EX. Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros for f (x) = −x^4 − 3x^3 + 6x^2 − 4x − 12.
Try It # Use Descartes’ Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for f (x) = 2x^4 − 10x^3 + 11x^2 − 15x + 12. Then use a graph to verify the numbers of positive and negative real zeros for the function.
