Problem 47
Question
Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=2 x^{4}-x^{3}-x^{2}+2 x+5$$
Step-by-Step Solution
Verified Answer
There are up to 3 positive and 1 negative real roots for \(p(x) = 2x^4 - x^3 - x^2 + 2x + 5\).
1Step 1: Identify Sign Changes for Positive Roots
To find the number of positive real roots, we need to count the number of sign changes in \(p(x)\). A sign change happens whenever consecutive terms of \(p(x)\) are positive and then negative or negative and then positive. So, in \(p(x) = 2x^4 - x^3 - x^2 + 2x + 5\), there are 3 sign changes: \(2x^4\) to \(-x^3\), \(-x^3\) to \(-x^2\), and \(-x^2\) to \(2x\). So, there are up to 3 positive real roots.
2Step 2: Identify Sign Changes for Negative Roots
To find the number of negative roots, substitute \(x\) with \(-x\) in \(p(x)\) and then count the sign changes. The new function, \(p(-x) = 2(-x)^4 - (-x)^3 - (-x)^2 + 2(-x) + 5\), simplifies to \(p(-x) = 2x^4 + x^3 - x^2 - 2x + 5\). There is only one sign change in \(p(-x)\), from \(x^3\) to \(-x^2\). Therefore, there are up to 1 negative real root.
Key Concepts
Polynomial RootsSign ChangesPositive Real RootsNegative Real Roots
Polynomial Roots
Understanding polynomial roots is fundamental for anyone diving into algebra. Roots of a polynomial are the values of the variable, often represented as 'x', that make the polynomial equal to zero. To find these values is to solve the equation, which can often be visualized as the points where the graph of the polynomial intersects the x-axis.
When dealing with polynomials, they can have as many roots as their highest exponent suggests—a concept known as the Fundamental Theorem of Algebra. For instance, a polynomial with a degree of four will have up to four roots, which may be real or complex numbers. These roots can be distinct or repeated, but the polynomial will exhibit behavior consistent with having roots at certain points, affecting its graph's curvature and direction.
When dealing with polynomials, they can have as many roots as their highest exponent suggests—a concept known as the Fundamental Theorem of Algebra. For instance, a polynomial with a degree of four will have up to four roots, which may be real or complex numbers. These roots can be distinct or repeated, but the polynomial will exhibit behavior consistent with having roots at certain points, affecting its graph's curvature and direction.
Sign Changes
Sign changes in a polynomial, as used in Descartes' Rule of Signs, refers to the instances where consecutive coefficients change from positive to negative or vice versa. It's like a game of spotting flips in a sequence of plus and minus signs.
To identify these changes, we focus solely on the sign of each term in the polynomial when expressed in its standard form. In the polynomial given by Descartes' Rule of Signs, each sign change hints at the possibility of a real root. The rule then uses these changes to predict the maximum number of positive or negative real roots the polynomial may have. However, the actual number of roots may be exactly the maximum predicted number or less by a multiple of two.
To identify these changes, we focus solely on the sign of each term in the polynomial when expressed in its standard form. In the polynomial given by Descartes' Rule of Signs, each sign change hints at the possibility of a real root. The rule then uses these changes to predict the maximum number of positive or negative real roots the polynomial may have. However, the actual number of roots may be exactly the maximum predicted number or less by a multiple of two.
Positive Real Roots
The quest to determine the positive real roots of a polynomial using Descartes' Rule of Sign involves a cheerful drill: simply count the number of times the coefficients change from a positive to a negative value as we read from the highest degree term down to the constant term.
In our exercise, by inspecting each term of the polynomial and observing these changes, we learn the maximum number of times the graph could potentially cross the x-axis at points where x is positive. This aspect is critical as it sketches a part of the puzzle concerning where the solutions to the polynomial lie, offering valuable insights before any actual algebraic or numerical methods are employed to solve for the exact roots.
In our exercise, by inspecting each term of the polynomial and observing these changes, we learn the maximum number of times the graph could potentially cross the x-axis at points where x is positive. This aspect is critical as it sketches a part of the puzzle concerning where the solutions to the polynomial lie, offering valuable insights before any actual algebraic or numerical methods are employed to solve for the exact roots.
Negative Real Roots
Determining the number of negative real roots adds a twist to our sign-change tracking: we temporarily imagine every 'x' in the polynomial as '-x' and then reassess. This process is akin to walking through a mirror world where every left turn is now a right, changing our perspective but keeping the landscape familiar.
Descartes' Rule of Signs uses this 'mirror' trick to transform the polynomial and reveal the maximum number of sign changes for terms with a negative 'x'. We tabulate the sign flips exactly as before, but this time in the mirrored version of the polynomial. Once this is done, we have a reliable estimate of the maximum number of negative real roots, shedding light on the downward crossings of the x-axis in the graph of the polynomial.
Descartes' Rule of Signs uses this 'mirror' trick to transform the polynomial and reveal the maximum number of sign changes for terms with a negative 'x'. We tabulate the sign flips exactly as before, but this time in the mirrored version of the polynomial. Once this is done, we have a reliable estimate of the maximum number of negative real roots, shedding light on the downward crossings of the x-axis in the graph of the polynomial.
Other exercises in this chapter
Problem 46
Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree 5 ; zeros at -3 and 1 , ea
View solution Problem 47
A rectangular solid has a square base and a height that is 2 inches less than the length of one side of the base. What lengths of the base will produce a volume
View solution Problem 47
Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x-1}{2 x^{2}-5 x-3}$$
View solution Problem 47
This set of exercises will draw on the ideas presented in this section and your general math background. Explain why there cannot be two different points at whi
View solution