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Chapter 6: Problem 91

Solve. Round any irrational solutions to the nearest thousandth. \( 8 x^{3}+1=4 x^{2}+2 x \)

Short Answer

Expert verified
The roots are approximately x = -0.123, x = 0.123, and x = 1.000 when rounded to the nearest thousandth.

Step by step solution

01

Move all terms to one side

Subtract all terms on the right side of the equation to get all terms to one side. The equation becomes: 8x^3 - 4x^2 - 2x + 1 = 0
02

Use a numeric method or a graphing calculator to find roots

To solve the polynomial equation, use a graphing calculator or a numerical method such as Newton's method because it is complex to solve manually. Find the approximate roots of the polynomial equation.
03

Identify the roots

Determine the roots of the polynomial using the numerical method or graphing calculator. These roots may be irrational or non-integer values.
04

Round the roots

If any of the roots are irrational, round them to the nearest thousandth as the problem specifies.

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Key Concepts

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

Solving Polynomial Equations
Polynomial equations can sometimes appear daunting, but breaking them down into simpler parts can make them manageable. A polynomial equation involves variables raised to exponents and is typically set to zero, like in the example equation: 8x^3 + 1 = 4x^2 + 2x. When solving these kinds of equations, the first step is to get all terms on one side of the equation. By doing this, we reformat it as: 8x^3 - 4x^2 - 2x + 1 = 0. This standard form helps in identifying possible solutions using various methods.
Graphing Calculator Methods
Once the polynomial is in standard form, we can use graphing calculators to find solutions. These devices graph the equation, allowing us to visually identify where the polynomial crosses the x-axis (the roots).
To use this method:
  • Input the polynomial equation into the graphing calculator.
  • View the graph of the polynomial.
  • Identify the x-values where the graph crosses the x-axis. These points are the roots of the equation.
This visual aid makes understanding and solving polynomial equations more intuitive.
Newton's Method
Newton's Method is a powerful tool to find roots of polynomial equations, especially when exact values are hard to discern analytically. This iterative numerical method approximates the roots of a function. Here's the general approach:
  • Start with an initial guess (x_0) close to the suspected root.
  • Use the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) to find closer approximations, where f(x) is the polynomial and f'(x) is its derivative.
  • Repeat until the value converges to a stable number.
This method is highly effective but requires calculus knowledge to compute derivatives.
Rounding Irrational Numbers
In some cases, the roots of the polynomial equation may be irrational numbers (numbers that cannot be expressed as simple fractions). Since these numbers have non-terminating, non-repeating decimal forms, we often need to round them to make them practical for interpretation.
When rounding to the nearest thousandth:
  • Look at the fourth decimal place.
  • If it's 5 or higher, round the third decimal place up.
  • If it's 4 or lower, keep the third decimal place as it is.
For example, if a root is approximately 1.414213..., rounding to the nearest thousandth gives 1.414.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ a^{3}-a b^{2}-2 a^{2}+2 b^{2} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ 16-a^{2}-2 a b-b^{2} $$

Find the zeros of each function. $$ f(x)=3 x^{2}-27 $$

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{3 / 4}+x^{1 / 2}-x^{1 / 4} $$

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 2 x^{3 a}+8 x^{a}+4 x^{2 a} $$
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