Problem 30 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=2 x^{3}-3 x^{2}-2 x+3 $$

Step-by-Step Solution

Answer

The rational zeros of the polynomial are \( x = \frac{3}{2}, \ x = -1, \ x = -\frac{1}{2} \).

Step 1: State the Rational Root Theorem

To find the rational zeros of a polynomial, we use the Rational Root Theorem. This theorem states that any rational zero, expressed as \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.

Step 2: Identify Factors of Constant and Leading Coefficients

The constant term of \( P(x) = 2x^3 - 3x^2 - 2x + 3 \) is 3, and its factors are +1, -1, +3, -3. The leading coefficient is 2, and its factors are +1, -1, +2, -2.

Step 3: List All Possible Rational Zeros

Using the factors of the constant and leading coefficients, list all possible rational roots: \( \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2} \).

Step 4: Test Possible Zeros Using Synthetic Division

Use synthetic division to test each possible rational zero. Start with \( x = 1 \), \( x = -1 \), and so on. Once a zero is found, the polynomial is divided and the process is repeated on the resulting simpler polynomial until all zeros are found.

Step 5: Verify Rational Zeros

Use synthetic division or substitution to verify each zero. If the division has a remainder of zero or substitution results in zero, the tested number is a valid zero of the polynomial.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form \( x - c \) where \( c \) is a constant. This efficient technique simplifies the process compared to traditional polynomial long division. It's particularly helpful in evaluating possible rational zeros suggested by the Rational Root Theorem. To perform synthetic division:

• Write down the coefficients of the polynomial, including zeros for any missing terms.
• Choose a potential zero from your list of possibilities.
• Place this number outside the synthetic division setup.
• Bring down the leading coefficient to the bottom row.
• Multiply this number by the chosen potential zero and write the result under the next coefficient.
• Add the numbers in the column and write the result below.
• Repeat the process until all coefficients are used.

If synthetic division completes with a remainder of zero, the chosen value is a root of the polynomial. Thus, synthetic division not only tests possible zeros but also helps to simplify the polynomial further, revealing additional zeros.

Polynomial Zeros

Polynomial zeros, or roots, are the values of \( x \) that make the polynomial equal to zero. Finding these zeros is key to understanding the behavior and graph of the polynomial. The roots can be real or complex numbers. In the context of the Rational Root Theorem, we are interested in the rational zeros, which are values that are expressible as simple fractions.Understanding polynomial zeros gives insight into whether a polynomial can be factored and how it interacts with the x-axis:

• Each zero corresponds to an x-intercept on the graph of the polynomial.
• The number of zeros typically matches the highest degree of the polynomial, counting multiplicity.

To verify a rational zero, substitute it into the polynomial or use synthetic division. If either method results in zero, it confirms the value as a root.

Factors of Polynomial Coefficients

In the Rational Root Theorem, knowing the factors of the polynomial coefficients allows us to predict possible rational zeros. Specifically, the theorem states that a rational zero of the polynomial \( P(x) \) is in the form \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.For instance, consider the polynomial \( P(x) = 2x^{3} - 3x^{2} - 2x + 3 \):

• The constant term is 3, with factors \( \pm 1, \pm 3 \).
• The leading coefficient is 2, with factors \( \pm 1, \pm 2 \).

By combining these, we list all possible rational zeros as \( \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2} \). Testing these values using synthetic division or substitution will reveal which, if any, are actual zeros of the polynomial.