Problem 403 - Precalculus

Question

For the following exercises, use the Rational Zero Theorem to find all real zeros. $$ 2 x^{3}-3 x^{2}-x+1=0 $$

Step-by-Step Solution

Answer

The real zeros are \(x = 1\) and \(x = -\frac{1}{2}\).

Step 1: Identify the Coefficients and Constant

The given polynomial is \(2x^3 - 3x^2 - x + 1 = 0\). In this polynomial, the coefficients are \(2\), \(-3\), \(-1\) and the constant term is \(1\).

Step 2: List All Possible Rational Zeros

According to the Rational Zero Theorem, the possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. The factors of the constant \(1\) are \(\pm 1\), and the factors of the leading coefficient \(2\) are \(\pm 1, \pm 2\). This gives possible rational zeros: \(\pm 1, \pm \frac{1}{2}\).

Step 3: Test Possible Rational Zeros Using Synthetic Division

We test each possible rational zero using synthetic division. Start with \(x = 1\):- Coefficients: \(2, -3, -1, 1\).- Synthetic division with \(1\) yields a remainder of \(0\).This means \(x = 1\) is a root.

Step 4: Divide the Polynomial by \(x - 1\)

Since \(x = 1\) is a zero, divide \(2x^3 - 3x^2 - x + 1\) by \(x - 1\) to find the remaining quadratic factor. Using synthetic division, the quotient is \(2x^2 - x - 1\).

Step 5: Solve the Quadratic Equation

Solve \(2x^2 - x - 1 = 0\) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 2\), \(b = -1\), \(c = -1\). Calculate \(b^2 - 4ac = (-1)^2 - 4(2)(-1) = 9\). The solutions are:\[ x = \frac{-(-1) \pm \sqrt{9}}{4} = \frac{1 \pm 3}{4} \]This gives \(x = 1\) and \(x = -\frac{1}{2}\).

Step 6: List All Real Zeros

Combine the zeros found. The real zeros of the polynomial are \(x = 1\) and \(x = -\frac{1}{2}\), with \(x = 1\) having multiplicity 2.

Key Concepts

Polynomial Division

Polynomial division is a process used to divide two polynomials, similar to how you perform long division with numbers. This technique is essential for finding the roots of the polynomial, especially when you need to factor a polynomial completely.
In the case of our polynomial, given as \(2x^3 - 3x^2 - x + 1 = 0\), polynomial division helps us break it down into simpler factors. The division results in a quotient and sometimes a remainder. When there's no remainder, it indicates that the divisor is a factor of the polynomial.

• Long Division of Polynomials: This method resembles numeric long division. Here, you arrange the divisor and dividend in descending powers of the variable and perform division steps.
• Synthetic Division: A shortcut to long division, especially when dividing by linear polynomials of the form \(x - c\).

Understanding polynomial division is vital because once a factor is identified, such as \(x - 1\) for this polynomial, it simplifies the equation, making it easier to find all zeros.

Synthetic Division

Synthetic division is a streamlined form of polynomial division, which is most useful when dividing by linear terms, such as \(x - c\). It simplifies calculations by focusing only on coefficients.
This process quickly checks potential roots given by the Rational Zero Theorem. In the example, synthetic division is used to test potential zeros like \(x = 1\). If the remainder is zero, \(x - c\) is a factor of the polynomial.

• Only coefficients are used in synthetic division, omitting variables and their exponents.
• The process involves setting up a division table with the potential zero and the polynomial coefficients.
• Through operations of multiplication and addition, it determines whether the test zero is indeed a root.

Synthetic division is especially efficient in finding possible rational zeros, speeding up the process of factorizations and root verification.

Quadratic Formula

Once the polynomial is reduced to a quadratic, the quadratic formula can be used to find the remaining roots. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
In our example, after dividing out \(x - 1\) from the original polynomial, the resulting quadratic is \(2x^2 - x - 1 = 0\).

• Identify the coefficients in your quadratic: \(a = 2\), \(b = -1\), and \(c = -1\).
• Calculate the discriminant \(b^2 - 4ac\). If it's positive, there are two distinct real roots.
• Use the quadratic formula to find each root, satisfying the remainder of the polynomial equation.

This method secures accurate solutions efficiently, especially when factoring is complex or not evident.

Real Zeros

Real zeros of a polynomial are the x-values that make the polynomial equal to zero. Finding these zeros is crucial as they indicate the x-intercepts of the polynomial's graph.
In the exercise, we determine the real zeros of \(2x^3 - 3x^2 - x + 1 = 0\) using the Rational Zero Theorem and division methods.

• The Rational Zero Theorem provides a list of potential rational zeros based on the constant term and leading coefficient.
• Testing these potential zeros through division leads us to the actual zeros.
• The zeros found in this example are \(x = 1\) and \(x = -\frac{1}{2}\), with \(x = 1\) occurring twice, also known as having a multiplicity of 2.

Recognizing real zeros is critical for understanding the polynomial's behavior and graph, offering insights into its characteristics, such as turning points and intercepts.