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Magazine Chapter 4: Problem 52
Solve. \(x^{3}-3 x^{2}-x+3=0\)
Short Answer
Expert verified
The roots are \(x = 1\), \(x = 3\), and \(x = -1\).
Step by step solution
01
Identify Possible Rational Roots
Use the Rational Root Theorem to list all possible rational roots of the polynomial \(x^3 - 3x^2 - x + 3 = 0\). The possible rational roots are the factors of the constant term (3) divided by the factors of the leading coefficient (1). The potential rational roots are \( \pm 1, \pm 3 \).
02
Test Rational Roots
Test each possible rational root using substitution into the polynomial. First, substitute \(x = 1\):\[1^3 - 3(1)^2 - 1 + 3 = 1 - 3 - 1 + 3 = 0\]Since this equals 0, \(x = 1\) is a root.
03
Divide the Polynomial
Using the root \(x = 1\), perform synthetic division or long division to divide \(x^3 - 3x^2 - x + 3\) by \(x - 1\). This will factor the polynomial:1. Write the coefficients: 1, -3, -1, 3.2. Bring down the leading coefficient (1).3. Multiply by the root 1, add to the next coefficient: repeat this step.The division yields: \(x^2 - 2x - 3\).
04
Solve the Quadratic Equation
Now, solve the quadratic equation \(x^2 - 2x - 3 = 0\) by factoring. Look for two numbers that multiply to -3 and add up to -2.The factors are \((x - 3)(x + 1)\) because:\[ (x - 3)(x + 1) = x^2 + 1x - 3x - 3 = x^2 - 2x - 3\]
05
Find All Roots
The quadratic factors \((x - 3)(x + 1) = 0\) give:- \(x - 3 = 0\Rightarrow x = 3\)- \(x + 1 = 0\Rightarrow x = -1\)Including the previously found root \(x = 1\), the roots of the original equation are \(x = 1, 3, -1\).
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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 powerful tool for polynomial equations. It helps us find potential rational roots of a polynomial function.
This theorem states that if a polynomial has a rational root expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are integers, then \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
This theorem states that if a polynomial has a rational root expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are integers, then \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.
- For the polynomial \(x^3 - 3x^2 - x + 3 = 0\), the constant term is 3 and the leading coefficient is 1.
- The factors of 3 are \(\pm 1, \pm 3\), and the only factor of 1 is \(\pm 1\).
Synthetic Division
Synthetic Division is a simplified method of polynomial division, especially useful when dividing by linear factors of the form \(x - c\). It is a streamlined algorithm that uses coefficients of the polynomial.
In our example, we need to divide \(x^3 - 3x^2 - x + 3\) by \(x - 1\) because we found \(x = 1\) as a root.
In our example, we need to divide \(x^3 - 3x^2 - x + 3\) by \(x - 1\) because we found \(x = 1\) as a root.
- Start by writing down the coefficients of the polynomial: 1, -3, -1, and 3.
- Remember: Bring the first coefficient down. Multiply this by the root \(1\), and add it to the next coefficient.
- Continue this multiplication and addition process to complete the division.
Quadratic Equations
Once a polynomial is reduced using Synthetic Division, it often leads to a quadratic equation.
In this case, we obtained \(x^2 - 2x - 3\), a simple quadratic to solve.
In this case, we obtained \(x^2 - 2x - 3\), a simple quadratic to solve.
- The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
- To solve quadratics, we can use methods such as factoring, completing the square, or the quadratic formula.
Factoring Polynomials
Factoring Polynomials involves breaking down a polynomial into simpler polynomials, making solving equations easier. It requires recognizing patterns and manipulating equations to find factors.
In this exercise, after synthetic division, we factor \(x^2 - 2x - 3\) by:
Thus, the original polynomial equation's roots are \(x = 1, x = 3, x = -1\), all found through thorough factorization and understanding of polynomial structures.
In this exercise, after synthetic division, we factor \(x^2 - 2x - 3\) by:
- Finding two numbers that multiply to -3 and add to -2: these numbers are -3 and 1.
- This gives us: \((x - 3)(x + 1)\) as factors.
Thus, the original polynomial equation's roots are \(x = 1, x = 3, x = -1\), all found through thorough factorization and understanding of polynomial structures.
