Lesson Objectives

- Learn how to find the possible number of positive real zeros
- Learn how to find the possible number of negative real zeros

## How to Use Descartes' Rule of Signs

In this lesson, we want to learn about Descartes' Rule of Signs. When trying to find the zeros for a polynomial function, Descartes' Rule of Signs can be helpful. It tells us the possible number of positive and negative real zeros for a polynomial function that meets the following conditions:

Example #1: Determine the possible number of positive and negative real zeros. $$f(x)=4x^5 - 2x^4 + 2x^3 - x^2 - 2x + 1$$ We will begin by thinking about the possible number of positive real zeros.

How many sign changes? Let's think about the coefficients:

Since there are 4 sign changes, we can conclude that there are either 4, 2, or 0 positive real zeros. Let's think about the possible number of negative real zeros: $$f(-x)=-4x^5 - 2x^4 - 2x^3 - x^2 + 2x + 1$$ How many sign changes? Let's think about the coefficients:

There is only 1 sign change in f(-x), which tells us there will be 1 negative real zero. We can't subtract 2 from 1 without going negative, so this is our final answer.

- Written in descending powers of x
- Has only real coefficients
- Has a nonzero constant term

### Descartes' Rule of Signs for Positive Zeros

- Count the number of sign changes that occur in the coefficients of the function
- The number of positive real zeros is equal to the number found above or is less than that number by some positive even integer
- Summary: find the number of sign changes, then we decrease this number by 2 until we get to zero or can't subtract away 2 without going negative

### Descartes' Rule of Signs for Negative Zeros

- Plug in (-x) for each occurrence of x, in other words, find f(-x)
- Count the number of sign changes that occur in the coefficients of the function
- The number of negative real zeros is equal to the number found above or is less than that number by some positive even integer
- Summary: find f(-x), then find the number of sign changes, then we decrease this number by 2 until we get to zero or can't subtract away 2 without going negative

Example #1: Determine the possible number of positive and negative real zeros. $$f(x)=4x^5 - 2x^4 + 2x^3 - x^2 - 2x + 1$$ We will begin by thinking about the possible number of positive real zeros.

How many sign changes? Let's think about the coefficients:

Coefficient | Change | Count |
---|---|---|

+4 | No | 0 |

-2 | Yes | 1 |

+2 | Yes | 2 |

-1 | Yes | 3 |

-2 | No | 3 |

+1 | Yes | 4 |

Coefficient | Change | Count |
---|---|---|

-4 | No | 0 |

-2 | No | 0 |

-2 | No | 0 |

-1 | No | 0 |

+2 | Yes | 1 |

+1 | No | 1 |

#### Skills Check:

Example #1

State the possible number of positive and negative real zeros. $$f(x)=27x^6 - 35x^3 + 8$$

Please choose the best answer.

A

+ zeros: 4, 2, or 0, - zeros: 3 or 1

B

+ zeros: 2 or 0, - zeros: 0

C

+ zeros: 0, - zeros: 1 or 3

D

+ zeros: 1 or 3, - zeros: 0

E

+ zeros: 2 or 0, - zeros: 0

Example #2

State the possible number of positive and negative real zeros. $$f(x)=5x^6 - 3x^4 - 80x^2 + 48$$

Please choose the best answer.

A

+ zeros: 0, - zeros: 3 or 1

B

+ zeros: 2 or 0, - zeros: 3 or 1

C

+ zeros: 4, 2, or 0, - zeros: 2 or 0

D

+ zeros: 2 or 0, - zeros: 2 or 0

E

+ zeros: 4, 2, or 0, - zeros: 3 or 1

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