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Descartes Rule of Signs

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Descartes Rule of Signs

Study Guide

Key Definition

Descartes' Rule of Signs states that the possible number of positive roots of a polynomial is equal to the number of sign changes in its nonzero coefficients, or less than that number by a non-negative multiple of 2 (so the result remains non-negative). When counting sign changes, skip any coefficients equal to 0.

Important Notes

Arrange the polynomial terms in descending order of exponents before applying the rule.
Count the sign changes to determine possible positive roots.
Subtract multiples of 2 from the sign changes for alternative positive root counts (ensuring the count stays ≥ 0).
This rule provides a possibility, not certainty, of the number of positive roots.
For negative roots, apply the rule to $f(-x)$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Each sign change in the coefficients indicates at least one real positive zero (a crossing of the x-axis). Complex roots occur in conjugate pairs, so reducing the count by 2 accounts for those possibilities. Hence the number of positive roots can drop by multiples of 2.

Remember

Descartes' Rule of Signs helps estimate the number of positive roots without solving the polynomial.

Quick Reference

Sign Changes:Count changes in sign of nonzero coefficients

Positive Roots:$\text{Sign changes} - 2k$ (where $k$ is a non-negative integer, and the result ≥ 0)

Understanding Descartes Rule of Signs

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Video explanation of this concept

Beginner

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Beginner Explanation

This rule helps determine the possible number of positive roots in a polynomial by counting sign changes in its coefficients.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

Determine the possible number of positive roots for the polynomial $x^3 - 2x^2 + x - 1$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a roller coaster track, and you need to determine if the polynomial $f(x) = x^4 - 3x^3 + 2x^2 - x + 1$ could represent the track's shape with multiple upward slopes.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the polynomial $-x^5 + x^4 - x^2 + x - 1$. Challenge yourself to find the possible number of positive roots.

Recap

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