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

Verified step by step guidance

1. Understand what "variations in sign" means: it refers to the number of times the signs of the coefficients of consecutive terms in the polynomial change when written in standard form.
2. Write down the polynomial function and list the signs of its coefficients in order: \(f(x) = 2x^5 + 3x^4 - 8x^3 - 5x + 6\); the coefficients are \(2\), \(3\), \(-8\), \(-5\), and \(6\) with signs \(+, +, -, -, +\) respectively.
3. Count the number of sign changes between consecutive coefficients: from \(+\) to \(+\) (no change), from \(+\) to \(-\) (one change), from \(-\) to \(-\) (no change), from \(-\) to \(+\) (second change). So, there are 2 sign changes.
4. Compare the counted number of sign changes (2) with the statement's claim (3): since they are not equal, the statement is false.
5. Explain why the statement is false: the polynomial has only 2 variations in sign, not 3, because the signs of the coefficients change only twice when moving from the highest degree term to the constant term.

Key Concepts

Polynomial Function and Degree

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients. The degree of the polynomial is the highest exponent of the variable, which determines the general shape and behavior of the graph. For example, ƒ(x) = 2x^5 + 3x^4 - 8x^3 - 5x + 6 is a fifth-degree polynomial.

Sign Variation of Polynomial Functions

Sign variation refers to the number of times the values of a polynomial change from positive to negative or vice versa as x varies. This is related to the number of real roots and the behavior of the function between those roots. Counting sign variations helps in understanding the number of times the graph crosses or touches the x-axis.

Descartes' Rule of Signs

Descartes' Rule of Signs is a method to estimate the number of positive and negative real roots of a polynomial by counting the number of sign changes in the sequence of its coefficients. The maximum number of positive real roots equals the number of sign variations, and the number of negative real roots is found by applying the rule to ƒ(-x).