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Magazine Chapter 10: Problem 55
Use the rational root theorem, Descartes' rule of signs, and the theorem on bounds as aids in finding all solutions to each equation. $$x^{3}+x+10=0$$
Short Answer
Expert verified
The solutions to the equation are \(x = -2, 1+2i\), and \(1-2i \).
Step by step solution
01
List Potential Rational Roots
According to the Rational Root Theorem, any possible rational root of the polynomial equation must be a factor of the constant term divided by a factor of the leading coefficient. For the equation \[x^{3} + x + 10 = 0\], the constant term is 10 and the leading coefficient is 1. The possible rational roots are thus the factors of 10: \[\text{Factors of 10}: \ \text{卤1, 卤2, 卤5, 卤10}\].
02
Apply Descartes' Rule of Signs
Descartes' Rule of Signs helps determine the number of positive and negative real roots. For positive roots, count the sign changes in \[x^{3} + x + 10\]: no sign change, which means 0 positive roots. For negative roots, substitute \[x \rightarrow -x\] in the equation: \[-x^{3} - x + 10\]There is 1 sign change (from \[-x^{3} \rightarrow -x\] and \[-x \rightarrow +10\]), meaning 1 negative root.
03
Test Potential Rational Roots
Evaluate the polynomial at each potential rational root to find which one is a solution. Testing \(x = -2\): \[(-2)^{3} + (-2) + 10 = -8 - 2 + 10 = 0\] So, \(x = -2\) is a root of the polynomial.
04
Perform Synthetic Division
Use synthetic division to divide the polynomial \(x^{3} + x + 10\) by \(x + 2\) to find the remaining polynomial: \[1 \quad 0 \quad 1 \quad 10 \ -2 \quad \downarrow \quad -2 \quad 4 \quad -10 \ 1 \quad -2 \quad 3 \quad 0\] So, the quotient is \(x^{2} - 2x + 5\).
05
Solve the Quadratic Equation
Solve the quadratic equation \(x^{2} - 2x + 5 = 0\) using the quadratic formula: \[x = \frac{2 \times 2 \times \bb{-4}}{2 \times 1} = \frac{2 \times \bb{-4}}{2 \times 1} \] find the discriminant \[4 - 20 = -16\]. Since the discriminant is less than 0, the solutions are complex: \[x = 1 \ \text{卤2i}.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Root Theorem
The Rational Root Theorem is a useful tool for finding potential rational solutions (roots) to polynomial equations. This theorem states that any rational root of a polynomial equation, with integer coefficients, is a fraction \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term (the term without a variable)
- \( q \) is a factor of the leading coefficient (the coefficient of the term with the highest degree)
Descartes' Rule of Signs
Descartes' Rule of Signs is a method to determine the number of positive and negative real roots of a polynomial. It is based on counting the number of sign changes in the polynomial's coefficients.
- For positive roots, look at the polynomial as is, and count the number of times the signs of the coefficients change.
- For negative roots, substitute \( x \) with \( -x \) in the polynomial and count the sign changes in the new polynomial.
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \( x - c \). It is especially useful for testing potential roots found by the Rational Root Theorem.
To perform synthetic division:
To perform synthetic division:
- Write down the coefficients of the polynomial.
- Bring down the leading coefficient.
- Multiply this coefficient by the root being tested (in our case, \( -2 \)), and write the result under the next coefficient.
- Add this result to the next coefficient and repeat the process for the entire polynomial.
Quadratic Formula
When a polynomial is reduced to a quadratic equation (of the form \( ax^2 + bx + c = 0 \)), the quadratic formula can be used to find its roots. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the quadratic equation \( x^2 - 2x + 5 = 0 \), we use the formula with \( a = 1 \), \( b = -2 \) and \( c = 5 \).
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the quadratic equation \( x^2 - 2x + 5 = 0 \), we use the formula with \( a = 1 \), \( b = -2 \) and \( c = 5 \).
- First, find the discriminant: \( b^2 - 4ac = (-2)^2 - 4(1)(5) = 4 - 20 = -16 \)
- Since the discriminant is negative (-16), the roots are complex numbers.
- The roots are: \( x = 1 \pm 2i \), where \( i \) is the imaginary unit.
