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Magazine Chapter 1: Problem 2
Solve the equation \(x^{3}-5 x^{2}-8 x+12=0\), given that the sum of two of the roots is \(7 .\)
Short Answer
Expert verified
The roots are \(-2, 6,\) and \(1\).
Step by step solution
01
Identify the Equation
The given equation is \(x^{3}-5 x^{2}-8 x+12=0\). Let's denote the roots as \(a, b,\) and \(c\).
02
Use the Sum and Product of Roots
For a cubic equation \(ax^3 + bx^2 + cx + d = 0\), the sum of the roots \(a + b + c\) is given by \(-\frac{b}{a}\). Given the equation \(x^3 - 5x^2 - 8x + 12 = 0\), \(-b/a = 5\). Hence, \(a + b + c = 5\).
03
Incorporate the Given Sum of Roots
We are given that the sum of two of the roots is 7. Assume \(a + b = 7\). Therefore, \(c = 5 - 7 = -2\).
04
Use the Factor Theorem
Since \(c = -2\) is a root, \(x + 2 = 0\) is a factor. Use polynomial division or synthetic division to factor \(x^3 - 5x^2 - 8x + 12\) by \(x + 2\).
05
Factor the Polynomial
Divide \(x^3 - 5x^2 - 8x + 12\) by \(x + 2\): 1. Divide \(x^3\) by \(x\) to get \(x^2\).2. Multiply \(x^2\) by \(x + 2\) to get \(x^3 + 2x^2\).3. Subtract from \(x^3 - 5x^2 - 8x + 12\) to get \(-7x^2 - 8x + 12\).4. Divide \(-7x^2\) by \(x\) to get \(-7x\).5. Multiply \(-7x\) by \(x + 2\) to get \(-7x^2 - 14x\).6. Subtract each term to get \(6x + 12\).7. Divide \(6x\) by \(x\) to get \(6\).8. Multiply \(6\) by \(x + 2\) to get \(6x + 12\).9. Subtract to get \(0\).Thus, \(x^3 - 5x^2 - 8x + 12 = (x + 2)(x^2 - 7x + 6)\).
06
Solve the Quadratic Equation
Solve the quadratic equation \(x^2 - 7x + 6 = 0\) by factoring: 1. It can be factored as \((x - 6)(x - 1) = 0\).Thus, the solutions to the quadratic equation are \(x = 6\) and \(x = 1\).
07
Determine All Roots
Combine all roots: The roots of the cubic equation are \(x = -2, x = 6, x = 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Division
Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. In this specific exercise, we are dividing the cubic polynomial \(x^3 - 5x^2 - 8x + 12\) by the linear polynomial \(x + 2\). The steps are as follows:
- First, divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\) to get \(x^2\).
- Multiply \(x^2\) by the entire divisor \(x + 2\), resulting in \(x^3 + 2x^2\).
- Subtract \(x^3 + 2x^2\) from \(x^3 - 5x^2 - 8x + 12\), leaving \(-7x^2 - 8x + 12\).
- Repeat this process: divide \(-7x^2\) by \(x\) to get \(-7x\), multiply \(-7x\) by the divisor \(x + 2\), and subtract.
- Keep going until the remainder is zero, indicating that the division is complete.
Sum and Product of Roots
The sum and product of the roots of a polynomial give significant insights into the structure of the polynomial. For a cubic polynomial of the form \(ax^3 + bx^2 + cx + d = 0\):
- The sum of the roots \(a + b + c\) is \(-\frac{b}{a}\).
- The sum of the product of the roots taken two at a time \(ab + ac + bc\) is \(\frac{c}{a}\).
- The product of the roots \(abc\) is \(-\frac{d}{a}\).
Factor Theorem
The factor theorem is a wonderful tool for finding the roots of a polynomial. It states that \(x - c\) is a factor of the polynomial \(f(x)\) if and only if \(f(c) = 0\). This means:
- If you substitute \(c\) into the polynomial and the result is zero, then \(c\) is a root, and \(x - c\) is a factor.
- In the given problem, since we deduced that one of the roots is -2, it follows that \(x + 2\) is a factor (because \(x - (-2)\) = \(x + 2\)).
Quadratic Factoring
Once a polynomial is reduced to a quadratic equation, quadratic factoring techniques can be used to find its roots. For example, the quadratic \(x^2 - 7x + 6\) needs to be factored. The steps are:
- Look for two numbers that multiply to the constant term (6) and add up to the coefficient of \(x\) (-7).
- In this case, these numbers are -6 and -1, because \(-6 \times -1 = 6\) and \(-6 + -1 = -7\).
- The quadratic equation \(x^2 - 7x + 6\) can thus be factored into \((x - 6)(x - 1)\).
