Select the correct answer from each drop-down menu.
Consider the equation below.
[tex]x^3-3 x^2-4=\frac{1}{x-1}+5[/tex]
The solutions to the equation are approximately [tex]x=0.91[/tex] and [tex]x=[/tex] ______ and [tex]x=[/tex] ______.
Answer (1)
Rewrite the given equation as a polynomial equation: x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0 .
Find the approximate real roots of the polynomial equation.
The approximate real roots are x ≈ 0.91 and x ≈ 3.69 .
The solutions to the equation are approximately 3.69 .
Explanation
Problem Analysis We are given the equation x 3 − 3 x 2 − 4 = x − 1 1 + 5 . Our goal is to find the approximate solutions for x .
Rewriting the Equation First, let's rewrite the equation by subtracting x − 1 1 and 5 from both sides: x 3 − 3 x 2 − 4 − x − 1 1 − 5 = 0 This simplifies to: x 3 − 3 x 2 − 9 − x − 1 1 = 0
Eliminating the Fraction To get rid of the fraction, we multiply both sides of the equation by ( x − 1 ) , assuming x = 1 : ( x − 1 ) ( x 3 − 3 x 2 − 9 ) − 1 = 0 Expanding this, we get: x 4 − 3 x 3 − 9 x − x 3 + 3 x 2 + 9 − 1 = 0 Which simplifies to: x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0
Defining the Polynomial Let f ( x ) = x 4 − 4 x 3 + 3 x 2 − 9 x + 8 . We are given that one approximate solution is x ≈ 0.91 . We need to find the other approximate real solutions.
Finding the Roots Using a calculator or software, we find the approximate real roots of the equation x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0 are approximately x ≈ 0.9067 and x ≈ 3.6888 .
Rounding the Roots Rounding to two decimal places, we have x ≈ 0.91 and x ≈ 3.69 .
Final Answer Therefore, the solutions to the equation are approximately x = 0.91 and x = 3.69 .
Examples
Polynomial equations like this can model various real-world phenomena, such as the trajectory of a projectile or the behavior of electrical circuits. Finding the roots of the polynomial helps us understand the critical points or stable states of the system. For example, in circuit analysis, the roots might represent the resonant frequencies of the circuit, which are crucial for designing filters and amplifiers. By solving such equations, engineers can optimize the performance and stability of these systems.
