Determine whether each statement is true or false. If false, explain why. The polynomial function ƒ(x)=2x^5+3x^4-8x^3-5x+6 has three variations in sign.

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = 2x^5 + 3x^4 - 8x^3 - 5x + 6 \)
2. List the coefficients of the polynomial: \( 2, 3, -8, -5, 6 \)
3. Examine the sign of each coefficient: \( 2 \) is positive, \( 3 \) is positive, \( -8 \) is negative, \( -5 \) is negative, \( 6 \) is positive
4. Count the number of sign changes between consecutive coefficients: from \( 3 \) to \( -8 \) (positive to negative), from \( -5 \) to \( 6 \) (negative to positive)
5. Conclude the number of sign variations: There are two sign changes, not three

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is ƒ(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where 'n' is a non-negative integer and 'a_n' are constants. Understanding the structure of polynomial functions is essential for analyzing their behavior, including their roots and end behavior.

Variations in Sign

Variations in sign refer to the changes in the sign of the coefficients of a polynomial as you evaluate it at different values of 'x'. Each time the polynomial changes from positive to negative or vice versa, it counts as a variation. The number of variations can provide insights into the number of positive and negative roots of the polynomial, which is crucial for determining its overall behavior.

Descartes' Rule of Signs

Descartes' Rule of Signs is a method used to determine the number of positive and negative real roots of a polynomial function based on the variations in sign of its coefficients. For positive roots, the number of sign changes in ƒ(x) gives the maximum number of positive roots, while for negative roots, the number of sign changes in ƒ(-x) indicates the maximum number of negative roots. This rule is fundamental for analyzing the roots of polynomials and understanding their graphical representation.