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Chapter 2: Problem 89

Using Descartes's Rule of Signs, use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{3}+3 x^{2}+1$$

Short Answer

Expert verified
There are 0 possible positive real zeros and 2 or 0 possible negative real zeros.

Step by step solution

01

Identify the Number of Sign Changes for Positive Real Zeros

The given polynomial function is \(h(x)=2 x^{3}+3 x^{2}+1\). Looking at the function, the coefficients of the polynomial do not change sign, so there are 0 sign changes. This means there are 0 possible positive real zeros.
02

Identify the Number of Sign Changes for Negative Real Zeros

Replace \(x\) with \(-x\) in the function to check the sign changes for the possibility of negative real zeros. Our new equation is \(h(-x)=2(-x)^{3}+3(-x)^{2}+1\), which simplifies to \(-2x^3 + 3x^2 +1\). As we can see, the sign changes twice in this equation, from positive to negative and from negative to positive, thus there can be 2 or 0 (2 - a multiple of 2) negative real zeros.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are mathematical expressions involving sums of powers of variables. In the simplest form, they include coefficients, variables, and exponents. A standard polynomial function can be written as:
  • \(a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0\)
Here, \(a_n, a_{n-1}, \ldots, a_0\) are constants known as coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial.
The example polynomial \(h(x)=2x^3+3x^2+1\) is a cubic polynomial because the highest power of \(x\) is three.
Polynomials are crucial in algebra because they represent a wide variety of functions and are used to solve diverse problems in mathematics and other scientific disciplines.
Real Zeros
Real zeros of a polynomial function are the values of \(x\) for which the polynomial equals zero. They are the solutions to the equation \(h(x) = 0\).
  • For example, if you have a polynomial like \(h(x)=2x^3+3x^2+1\), finding the real zeros involves solving \(2x^3+3x^2+1 = 0\).
Real zeros are significant because they represent the points where the graph of the polynomial touches or crosses the x-axis.
Identifying the number of real zeros helps understand the behavior of the polynomial's graph and the relationship between roots and coefficients.
Sign Changes
Sign changes in the context of Descartes's Rule of Signs refer to changes in the signs of consecutive coefficients in a polynomial when listed in standard form.
  • For instance, in the polynomial \(2x^3+3x^2+1\), there are no sign changes because all the coefficients are positive.
When determining zeros, Descartes鈥檚 Rule of Signs uses sign changes to predict the number of possible positive and negative roots.
It's essential to notice if there are sign changes, as they indicate potential transitions of the graph of the polynomial on the x-axis.
Positive and Negative Roots
Determining the number of positive and negative roots of a polynomial function is pivotal in understanding its zeroes and overall behavior. Descartes's Rule of Signs states:
  • The number of positive roots is equal to the number of sign changes in the original polynomial, or less by an even number.
  • The number of negative roots is calculated by replacing \(x\) with \(-x\) and counting the sign changes, then adjusting by even numbers.
For instance, in \(h(x)=2x^3+3x^2+1\), there are no sign changes, indicating zero positive roots.
However, substituting \(-x\) into the polynomial yields \(-2x^3 + 3x^2 +1\), which has two sign changes, suggesting two or zero negative roots.
This allows predictions about the behavior of function graphs and the real zero locations.

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Most popular questions from this chapter

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