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Magazine Chapter 5: Problem 32
For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}-3 x^{2}+4 x+3=0 $$
Short Answer
Expert verified
The real solution is \( x = \frac{1}{2} \). The quadratic has no real solutions.
Step by step solution
01
Identify Possible Rational Zeros
According to the Rational Zero Theorem, any rational solution of a polynomial equation \( a_nx^n + a_{n-1}x^{n-1} + ... + a_0 = 0 \) must be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \). For the equation \( 2x^3 - 3x^2 + 4x + 3 = 0 \), the constant term is 3, and the leading coefficient is 2. The factors of 3 are \( \pm 1, \pm 3 \), and the factors of 2 are \( \pm 1, \pm 2 \). Possible rational zeros are \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \).
02
Test Possible Zeros
Substitute each possible rational zero into the polynomial equation to determine if it is indeed a zero. \( f(x) = 2x^3 - 3x^2 + 4x + 3 \). Start by testing \( x = 1 \):### Calculation\( f(1) = 2(1)^3 - 3(1)^2 + 4(1) + 3 = 2 - 3 + 4 + 3 = 6 \)### ConclusionSince \( f(1) eq 0 \), \( x = 1 \) is not a zero. Continue this process with each possible rational zero.
03
Finding the Correct Zero
Continue testing each potential zero:1. Test \( x = -1 \): \[ f(-1) = 2(-1)^3 - 3(-1)^2 + 4(-1) + 3 = -2 - 3 - 4 + 3 = -6 \] \( f(-1) eq 0 \). Thus, \( x = -1 \) is not a zero.2. Test \( x = \frac{1}{2} \): \[ f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right)^2 + 4\left(\frac{1}{2}\right) + 3 = \frac{2}{8} - \frac{3}{4} + 2 + 3 = 0 \] \( f\left(\frac{1}{2}\right) = 0 \). Thus, \( x = \frac{1}{2} \) is a solution.
04
Polynomial Division
Since \( x = \frac{1}{2} \) is a zero, divide the polynomial by \( 2x - 1 \) using synthetic division to find the remaining quadratic. Performing the division:### Calculation- Coefficients: 2, -3, 4, 3- Use \( \frac{1}{2} \) for synthetic division: - Drop 2 down - Multiply \( 2 \times \frac{1}{2} = 1 \), add to -3 gives -2 - Multiply \( -2 \times \frac{1}{2} = -1 \), add to 4 gives 3 - Multiply \( 3 \times \frac{1}{2} = \frac{3}{2} \), add to 3 gives 0 confirming division success since remainder is 0.- The quotient polynomial is \( 2x^2 - 2x + 3 \).
05
Solving the Quadratic
Find the roots of the quadratic equation \( 2x^2 - 2x + 3 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -2 \), and \( c = 3 \).### Calculation\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 2 \times 3}}{2 \times 2}\]\[x = \frac{2 \pm \sqrt{4 - 24}}{4} = \frac{2 \pm \sqrt{-20}}{4} \]Because the discriminant \(-20\) is negative, the quadratic has no real solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Equations
Polynomial equations are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A general polynomial equation in one variable looks like this:
- The highest power, known as the degree of the polynomial, determines the behavior and complexity of the equation.
- Coefficients and constant terms are real numbers that influence the solution set and nature of the polynomial.
- In our example, the polynomial equation is given as: \[ 2x^3 - 3x^2 + 4x + 3 = 0 \]Having a degree of 3, which means it's a cubic equation.
Synthetic Division
Synthetic division is a simplified form of polynomial division, particularly useful when dividing a polynomial by a linear factor of the form \( x - c \). It's an efficient way to determine if a given number is a zero of the polynomial, which is to say it divides the polynomial without a remainder. The steps involved are:
- Write down the coefficients of the polynomial.
- Use the potential zero as the divisor.
- Perform operations to determine if there's a remainder.
Quadratic Formula
The quadratic formula is a reliable tool for solving quadratic equations, which are second-degree polynomials of the form \( ax^2 + bx + c = 0 \). The formula given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Allows you to find solutions by computing specific values:
- \( a \), \( b \), and \( c \) are coefficients from the polynomial.
- The "plus-minus" symbol \( \pm \) indicates there can be two solutions based on the sign, which are called roots of the equation.
Discriminant
The discriminant is a component of the quadratic formula, given by \( b^2 - 4ac \), that provides insights into the nature of the roots of a quadratic equation without actually solving it. Here's why it matters:
- If the discriminant is positive, there are two distinct real roots.
- If it's zero, there is exactly one real root, meaning the quadratic "touches" the x-axis.
- A negative discriminant, as seen in \( 2x^2 - 2x + 3 \) that had a discriminant of \(-20\), means no real roots exist, only complex ones are possible.
