4. Polynomial and Rational Functions

Use Rational Zero Theorem to Find Possible Rational Zeros

Zeros of Polynomial Functions

The Rational Zeros Theorem

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91

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

Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x^4−3x^2−4). If f(3)=−150, determine the value of a_n.

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 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)=3x^4−11x^3−x^2+19x+6

Exercises 53–60 show incomplete graphs of given polynomial functions.  a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x^5+2x^4−15x^3−10x^2+12x+8

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x^3+x^2−3x+1

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x^3−3x^2−11x+6

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

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

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. 2x^3−x^2−9x−4=0

In Exercises 33–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x^4−5x^3−x^2−6x+4

Solving Polynomial Equations By Factoring and Using Synthetic Division

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−x^2+19x+6

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

Exercises 53–60 show incomplete graphs of given polynomial functions.  a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=−x^3+x^2+16x−16

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; -2, 5, and 3+2i are zeros; f(1) = -96

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^5−x^4−7x^3+7x^2−12x−12

Find the average rate of change of f(x)=√x from x1=4 to x2=9.

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x^3−2x^2−11x+12=0

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

Finding the Equation of a Polynomial from a Graph

Exercises 53–60 show incomplete graphs of given polynomial functions.  a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=4x^3−8x^2−3x+9

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

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. x^4−3x^3−20x^2−24x−8=0

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^3−4x^2−7x+10

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. 2x^5+7x^4−18x^2−8x+8=0

In Exercises 25–32, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; i and 3i are zeros; f(-1) = 20

Pre-Calculus - Using Descartes rule of signs

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x^4−x^3+5x^2−2x−6

Ex: The Rational Root (Zero) Theorem

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

In Exercises 33–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x^3−3x^2+3x−1

In Exercises 33–38, use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x^3+2x^2+5x+4

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. 4x^4−x^3+5x^2−2x−6=0

Finding all the Zeros of a Polynomial - Example 1

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x+6=0

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

How To Find The Real & Imaginary Solutions of Polynomial Equations

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3−2x^2−11x+12

Determine the equation of a polynomial by analyzing its graph

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3+4x^2−3x−6

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

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/dda4f49dc11fe8eb-1672428750420.jpg" alt="" /></p>

Finding the Zeros of a Polynomial from Start to Finish

Exercises 53–60 show incomplete graphs of given polynomial functions.  a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x^4−3x^3−7x^2−8x+6

Find the Zeros of a Polynomial Function with Imaginary Zeros

Rational Root Theorem

Find all zeros of f(x) = x³ + 5x² – 8x + 2.

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−3x^2−6x+8

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x^3−10x−12=0

Finding All Zeros of a Polynomial Equation

Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x−1=0

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^3+x^2−4x−4

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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College Algebra

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Use Long Division to Divide Polynomials

Use Synthetic Division to Divide Polynomials

Evaluate a Polynomial Using the Remainder Theorem

Use the Factor Theorem to Solve a Polynomial Equation

Dividing Polynomials

Identify Polynomial Functions

Recognize Characteristics of Graphs of Polynomial Functions

Determine End Behavior

Use Factoring to Find Zeros of Polynomial Functions