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Magazine Chapter 8: Problem 82
Solve $$ 10 x^{3}+10 x-30 x^{2}-30=0 $$
Short Answer
Expert verified
The solutions are \(x = 3, x = i, x = -i\).
Step by step solution
01
Simplify the Equation
Start by factoring out the greatest common factor from the entire equation. In this case, the greatest common factor is 10:\[10(x^3 + x - 3x^2 - 3) = 0\]Divide the entire equation by 10 to simplify:\[x^3 + x - 3x^2 - 3 = 0\]
02
Rearrange Terms
Rearrange the terms to organize the polynomial more clearly:\[x^3 - 3x^2 + x - 3 = 0\]
03
Factor the Polynomial
Look for possible ways to factor the polynomial. Notice that we can group terms and factor by grouping:\[(x^3 - 3x^2) + (x - 3)\]Factor out the common factors in each group:\[x^2(x - 3) + 1(x - 3)\]This can be factored further as:\[(x^2 + 1)(x - 3) = 0\]
04
Solve the Factored Equation
Set each factor from \[(x^2 + 1)(x - 3) = 0\] equal to zero and solve for \(x\):1. \(x^2 + 1 = 0\): \[x^2 = -1\] \[x = \,\pm i\] (where \(i\) is the imaginary unit)2. \(x - 3 = 0\): \[x = 3\]
05
Verify the Solutions
Substitute the solutions \(x = 3\) and \(x = \pm i\) back into the original equation to verify:1. For \(x = 3\), substitute into the original equation: - \[x = 3\] satisfies the equation because the original equation simplifies to zero.2. For \(x = i\) or \(x = -i\), the substitution shows the left side of the equation simplifies correctly for complex solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Equations
When tackling mathematical problems, polynomial equations are a frequent encounter. These equations involve expressions where the highest degree of variable exponent defines the order of the polynomial. For instance, the given equation is a third-degree polynomial because the highest power of the variable, here denoted as \(x^3\), is 3. In a polynomial equation like \(10x^3 + 10x - 30x^2 - 30 = 0\), components include coefficients and terms that may be arranged in any sequence, making comprehension crucial.
- The degree indicates the highest power of the variable.
- Coefficients are the numbers in front of the variable terms.
- Operations within polynomial equations typically involve addition, subtraction, and multiplication.
Imaginary Numbers
Imaginary numbers introduce a fascinating aspect to solving equations, particularly those where real solutions are not possible due to negative under-root values. In mathematics, the imaginary unit denoted by \(i\) is defined as \(\sqrt{-1}\). This unit facilitates easier handling of scenarios that result in negative square roots. For instance, in our factored polynomial equation, one of the parts, \(x^2 + 1 = 0\), resolves into imaginary numbers because
- Solving \(x^2 = -1\) gives \(x = \pm \sqrt{-1}\).
- This translates to \(x = \pm i\).
Solving Polynomials
Solving polynomials requires a systematic approach that varies with the degree and form of the polynomial. Here, our goal is to find the values of \(x\) that satisfy the polynomial equation \(x^3 + x - 3x^2 - 3 = 0\). This process oftentimes involves several strategic steps:
- First, simplify the equation by factoring out the greatest common factor.
- Next, organize the equation to make spotting factoring patterns easier.
- Use techniques like grouping to factor simpler quadratic equations within the polynomial.
