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Magazine Chapter 3: Problem 700
For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation. $$3 x^{3}+11 x^{2}+8 x-4=0$$
Short Answer
Expert verified
The solutions are \(x = -2\) and \(x = \frac{1}{3}\).
Step by step solution
01
Identify Potential Rational Zeros
According to the Rational Zero Theorem, the possible rational zeros of a polynomial function are the ratios of the factors of the constant term to the factors of the leading coefficient. The constant term in the polynomial \( 3x^3 + 11x^2 + 8x - 4 \) is \(-4\) and the leading coefficient is \(3\). The factors of \(-4\) are \( \pm 1, \pm 2, \pm 4 \), and the factors of \(3\) are \( \pm 1, \pm 3 \). Therefore, the potential rational zeros are: \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3} \).
02
Test Potential Zeros Using Synthetic Division
To find the actual zero, we use synthetic division with each of the potential rational zeros. We start by testing \( x = 1 \). Setting up synthetic division with 1, the coefficients from the polynomial \(3, 11, 8, -4\) are used:\[\begin{array}{r|rrrr}1 & 3 & 11 & 8 & -4 \ & & 3 & 14 & 22 \\hline & 3 & 14 & 22 & 18\end{array}\]Since the remainder is 18, \( x = 1 \) is not a zero. Repeat this with other values until the remainder is 0.
03
Find a Successful Zero
Using synthetic division with \( x = -1 \), perform the following:\[\begin{array}{r|rrrr}-1 & 3 & 11 & 8 & -4 \ & & -3 & -8 & 0 \\hline & 3 & 8 & 0 & -4\end{array}\]Here, the remainder is -4, so \( x = -1 \) is not a zero. Testing \( x = 2 \):\[\begin{array}{r|rrrr}2 & 3 & 11 & 8 & -4 \ & & 6 & 34 & 84 \\hline & 3 & 17 & 42 & 80 \end{array}\]The remainder is not zero. Finally, test \( x = -2 \):\[\begin{array}{r|rrrr}-2 & 3 & 11 & 8 & -4 \ & & -6 & -10 & 4 \\hline & 3 & 5 & -2 & 0\end{array}\]This time, the remainder is 0, indicating \( x = -2 \) is a root.
04
Factor and Solve the Remaining Polynomial
Since \( x = -2 \) is a factor, use synthetic division to reduce the polynomial to \(3x^2 + 5x - 2\). Solve \(3x^2 + 5x - 2 = 0\) by the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \(a = 3\), \(b = 5\), \(c = -2\):\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} \]\[ x = \frac{-5 \pm \sqrt{25 + 24}}{6} \]\[ x = \frac{-5 \pm \sqrt{49}}{6} \]\[ x = \frac{-5 \pm 7}{6} \]The solutions are \( x = \frac{2}{6} = \frac{1}{3} \) and \( x = \frac{-12}{6} = -2 \).
05
List All Solutions
The solutions to the polynomial equation \( 3x^3 + 11x^2 + 8x - 4 = 0 \) are \( x = -2, \frac{1}{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Synthetic Division
Synthetic division is a simplified form of dividing a polynomial, where you reduce the original polynomial by a potential zero. This method is highly efficient when the divisor is a linear polynomial of the form \(x - c\). The process allows you to check whether a particular value is a zero of the polynomial.
Here's a quick rundown on how it works:
Here's a quick rundown on how it works:
- Write down the coefficients of the polynomial in order.
- Choose a potential rational zero and write it to the side.
- Bring down the first coefficient directly.
- Multiply this coefficient by the potential zero and add it to the next coefficient.
- Repeat the process for all coefficients.
Polynomial Equation
A polynomial equation is an expression set to zero that consists of variables raised to whole number exponents and their coefficients. In our exercise, the polynomial equation is \(3x^3 + 11x^2 + 8x - 4 = 0\). Solving it means finding all values of \(x\) that make the equation true.
Polynomial equations can have:
Polynomial equations can have:
- Real or complex solutions.
- Multiple solutions, where solutions can be repeated.
- Solutions derived through various techniques such as factoring, synthetic division, or the quadratic formula.
Quadratic Formula
Once a polynomial is reduced, sometimes down to a quadratic form through techniques like synthetic division, we can apply the quadratic formula to find its roots. This formula is a go-to tool when a polynomial can be simplified to \(ax^2 + bx + c = 0\).
The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works:
The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works:
- Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic expression.
- Plug these values into the formula.
- Solve under the square root, known as the discriminant \(b^2 - 4ac\), to determine the nature and number of solutions.
- Compute the solutions by performing the operations indicated in the formula.
