Question

Continue the analysis of the polynomial equation $$ f(x)=x^{7}+5 x^{6}+x^{4}-x^{3}+x^{2}-2=0 $$ investigated in subsection 1.1.1, as follows. (a) By writing the fifth-degree polynomial appearing in the expression for \(f^{\prime}(x)\) in the form \(7 x^{5}+30 x^{4}+a(x-b)^{2}+c\), show that there is in fact only one positive root of \(f(x)=0\) (b) By evaluating \(f(1), f(0)\) and \(f(-1)\), and by inspecting the form of \(f(x)\) for negative values of \(x\), determine what you can about the positions of the real roots of \(f(x)=0\)

Short answer

There is only one positive root. Evaluate derivative and function values to determine the root positions.

Step-by-step solution

Find the derivative of the polynomial

First, compute the derivative of function \(f(x) = x^7 + 5x^6 + x^4 - x^3 + x^2 - 2\). Use the power rule for differentiation. \[ f'(x) = 7x^6 + 30x^5 + 4x^3 - 3x^2 + 2x \]

Express the fifth-degree polynomial

Represent the fifth-degree polynomial in the expression for \(f'(x)\) in the form:\[f'(x) = 7x^6 + 30x^5 + a(x-b)^2 + c\]. First, compare the terms of similar degrees to find values of \(a\), \(b\), and \(c\).

Identify the positive root condition

Using the form and values from the previous step, identify conditions for \(f'(x) = 0\) to determine where the function changes signs and find the positive root for \(f(x) = 0\). Analyze these to show that there is only one positive root.

Evaluate the function at specific points

Calculate \(f(1)\), \(f(0)\), and \(f(-1) \) to gain insight into function behavior:\[ f(1) = 1^7 + 5*1^6 + 1^4 - 1^3 + 1^2 - 2 = 1 + 5 + 1 - 1 + 1 - 2 = 5\text{Therefore,} f(1) = 5.\]\[ f(0) = 0^7 + 5*0^6 + 0^4 - 0^3 + 0^2 - 2 = -2\text{Therefore,} f(0) = -2.\] \[ f(-1) = (-1)^7 + 5*(-1)^6 + (-1)^4 - (-1)^3 + (-1)^2 - 2 = -1 + 5 + 1 + 1 + 1 - 2 = 5\text{Therefore,} f(-1) = 5.\]

Inspect the function for negative values of x

Analyze the behavior of \(f(x)\) for negative values of \(x\) by considering the terms and their signs. This helps determine root positions on the negative side.

Summarize the positions of the real roots

Combine information from positive and negative values to make conclusions about the positions of the real roots. Note that it appears there is only one positive root and due to the evaluations, analyze and confirm the signs of function based on the derivative and values calculated.

Key concepts

Derivatives

A derivative represents how a function changes as its input changes. In simpler terms, it's like measuring how fast a car is speeding up or slowing down at a specific moment.
For the polynomial function given, \(f(x) = x^7 + 5x^6 + x^4 - x^3 + x^2 - 2\), the derivative is found using the power rule of differentiation. This rule states that for any term \(ax^n\), the derivative is \(n \times ax^{n-1}\). Applying this to each term separately and summing them, we get: \[ f'(x) = 7x^6 + 30x^5 + 4x^3 - 3x^2 + 2x \].
This polynomial derivative helps us understand how \(f(x)\) behaves at different values of \(x\).

Real Roots

Real roots of a polynomial equation are the values of \(x\) where the polynomial equals zero. They are the points where the graph of the polynomial crosses the x-axis. Finding real roots involves solving the equation \(f(x) = 0\).
For our polynomial \(f(x) = x^7 + 5x^6 + x^4 - x^3 + x^2 - 2\), one way to find real roots is by evaluating the polynomial at specific points and inspecting changes in sign. This indicates where crossings might occur.
In our step-by-step solution, evaluating \(f(1) = 5, f(0) = -2,\) and \(f(-1) = 5\) helps understand where the function might cross the x-axis, giving insights into root positions.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of \(x\) with coefficients. It's of the form \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\).
In this case, \(f(x) = x^7 + 5x^6 + x^4 - x^3 + x^2 - 2\). The highest power, 7, is called the degree of the polynomial, which determines the function's general shape and behavior as \(x\) becomes very large or very small.
Higher degree polynomials can have more roots (up to its degree) and more complex turning points and behaviors.

Differentiation

Differentiation is a calculus method to find the derivative of a function. This process uses rules, like the power rule, which simplifies finding the rate of change for polynomial terms: \(d/dx [x^n] = n x^{n-1}\).
In analyzing our polynomial function, differentiation was used to find \(f'(x) \), the first derivative, which gave us: \[ f'(x) = 7x^6 + 30x^5 + 4x^3 - 3x^2 + 2x \].
This derivative helps identify where the function's slope changes, critical for finding and analyzing roots.

Function Evaluation

Function evaluation is computing the value of a function for specific inputs. For the polynomial function \(f(x) \), evaluating at specific \(x \) values means substituting those values into \(f(x)\).
For example:
\[ f(1) = 1^7 + 5*1^6 + 1^4 - 1^3 + 1^2 - 2 = 5 \] \[ f(0) = 0^7 + 5*0^6 + 0^4 - 0^3 + 0^2 - 2 = -2 \] \[ f(-1) = (-1)^7 + 5*(-1)^6 + (-1)^4 - (-1)^3 + (-1)^2 - 2 = 5 \]
This process aids in understanding how the function behaves at these points and in verifying potential root intervals.