4. Polynomial and Rational Functions

Use Rational Zero Theorem to Find Possible Rational Zeros

Zeros of Polynomial Functions

Ex: The Rational Root (Zero) Theorem

Finding all the Zeros of a Polynomial - Example 2

For Exercises 40–46,

(a) List all possible rational roots or rational zeros.
(b) Use Descartes’s Rule of Signs to determine the possible number of positive and negative real roots or real zeros.
(c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero.
(d) Use the quotient from part (c) to find all the remaining roots or zeros.


f(x) = x^3 + 3x^2 - 4

Find the Equation from the Graph of a Polynomial

Descartes' Rule of Signs

Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 2.

Use descartes rule of signs to find the number of positive and negative real zeros

Write the equation of a polynomial using its x-intercepts

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

The Rational Zeros Theorem

Solving Polynomial Equations By Factoring and Using Synthetic Division

Use Descartes' Rule of Signs to explain why 2x^4 + 6x^2 + 8 = 0 has no real roots.

Finding the Equation of a Polynomial from a Graph

Rational Roots Test / Theorem

Finding all the Zeros of a Polynomial - Example 3

In Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x) = -2(x - 1)(x + 2)^2(x+5)^2

Pre-Calculus - Using Descartes rule of signs

In Exercises 47–48, find an nth-degree polynomial function with real coefficients satisfying the given conditions. Verify the real zeros and the given function value. n = 3; 2 and 2 - 3i are zeros; f(1) = -10

Finding all the Zeros of a Polynomial - Example 1

Solving Higher Degree Polynomials by Synthetic Division and the Rational Roots Test

How To Find The Real & Imaginary Solutions of Polynomial Equations

Determine the equation of a polynomial by analyzing its graph

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. g(x) = x^4 - 6x^3 + x^2 + 24x + 16

Finding the Zeros of a Polynomial from Start to Finish

Find the Zeros of a Polynomial Function with Imaginary Zeros

Rational Root Theorem

Finding All Zeros of a Polynomial Equation

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. f(x) = 2x^4 + 3x^3 + 3x - 2

In Exercises 37–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x) = 3x^4 - 2x^3 - 8x + 5

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