Problem 17 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-3 x-2 $$

Step-by-Step Solution

Answer

Rational zeros are \(-1\) and \(2\); factored form is \((x + 1)^2(x - 2)\).

Step 1: Use the Rational Root Theorem

The Rational Root Theorem states that any potential rational root of a polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_0 = 0\) is of the form \(\frac{p}{q}\), where \(p\) divides the constant term \(a_0\) and \(q\) divides the leading coefficient \(a_n\). For \(P(x) = x^3 - 3x - 2\), the constant term is -2 and the leading coefficient is 1, so any rational root is \(\pm 1\) or \(\pm 2\).

Step 2: Test Rational Roots

Test the potential rational roots \(\pm 1\) and \(\pm 2\) to see if they are actually roots of the polynomial \(P(x) = x^3 - 3x - 2\).- For \(x = 1\): \(P(1) = 1^3 - 3 \times 1 - 2 = 1 - 3 - 2 = -4\) (not a root)- For \(x = -1\): \(P(-1) = (-1)^3 - 3 \times (-1) - 2 = -1 + 3 - 2 = 0\) (a root)- For \(x = 2\): \(P(2) = 2^3 - 3 \times 2 - 2 = 8 - 6 - 2 = 0\) (a root)- For \(x = -2\): \(P(-2) = (-2)^3 - 3 \times (-2) - 2 = -8 + 6 - 2 = -4\) (not a root)Thus, \(x = -1\) and \(x = 2\) are roots of the polynomial.

Step 3: Polynomial Division

Since \(x = -1\) and \(x = 2\) are roots, divide \(P(x)\) by \((x + 1)(x - 2)\). We can perform this step one root at a time using synthetic division.Perform synthetic division using \(x + 1\):- Coefficients: 1, 0, -3, -2- Result: \([1, -1, -2, 0]\), giving \(x^2 - x - 2\).Now, factor \(x^2 - x - 2\):- \(x^2 - x - 2 = (x - 2)(x + 1)\) (factoring by inspection).The complete factored form of \(P(x)\) is \((x + 1)^2(x - 2)\).

Key Concepts

Polynomial Division

Polynomial division is the process of dividing a polynomial by another polynomial of a lower degree. This is an important skill in algebra, allowing us to simplify complex expressions and find factors of a polynomial.

In the case of our polynomial \( P(x) = x^3 - 3x - 2 \), the division helps us verify the factors of the polynomial once we identify possible roots. We can use different methods, like long division or synthetic division, to achieve this.

Let's break down polynomial division into more digestible parts:

• Identify what needs to be divided: You start with the full polynomial, and you try to divide it by a simpler expression, usually in the form of \( (x - r) \), where \( r \) is a known root of the polynomial.
• Follow a systematic approach: In polynomial long division, we divide the terms in descending order of power, just like in traditional division. The remainder can help verify the accuracy of your factorization.
• Correct interpretation: Successful division confirms the divisibility of the polynomial, supporting the roots identified by the Rational Root Theorem.

Synthetic Division

Synthetic division offers a simpler and faster way to divide polynomials, especially when dividing by linear expressions of the form \((x - r)\). This method is particularly handy when using the roots found through the Rational Root Theorem.

Here's how synthetic division works:

• Write down the coefficients of the polynomial, ignoring any missing terms (adding zero coefficients if necessary).
• Use the root value \( r \) as a divisor. Start the process by bringing the initial coefficient down.
• Multiply this result by \( r \) and write the product underneath the next coefficient.
• Add down the column and repeat the multiply-and-add steps until completed.
• The final row of numbers represents your result: the last number is the remainder, while the others are coefficients of the quotient polynomial.

In our exercise, after finding the roots \( x = -1 \) and \( x = 2 \), we perform synthetic division first by \( (x + 1) \), achieving a result of \( x^2 - x - 2 \). This confirms \( x = -1 \) as a factor, making further solutions much easier.

Factored Form

A polynomial is in its factored form when it is expressed as a product of its factors, which are simpler polynomials or linear expressions. Factoring a polynomial reveals its roots and simplifies solving equations and graphing functions.

For \( P(x) = x^3 - 3x - 2 \), once we determined the roots \( x = -1 \) and \( x = 2 \), we used polynomial division to factor it further.

• Start with one known factor, like \( (x + 1) \), and divide it from the polynomial.
• Synthetic division confirms whether the factor divides the polynomial without a remainder.
• Continue to factor the quotient polynomial further, if possible.
• The full factored form, \( (x + 1)^2(x - 2) \), reveals that each bracket represents a root.

This form is particularly useful in identifying and verifying the solutions or roots of the polynomial efficiently. It also aids in graphing the function, as the zeros (roots) are immediately visible.

Polynomial Roots

The roots of a polynomial are the values for which the polynomial evaluates to zero. In simpler terms, these are the solutions to the polynomial equation \( P(x) = 0 \). Finding these solutions is a fundamental task in algebra.

In our problem, the Rational Root Theorem helped us identify the potential rational roots: \( \pm 1 \) and \( \pm 2 \). By testing each, we determined that \( x = -1 \) and \( x = 2 \) were, in fact, roots of our polynomial.

Here's what you need to understand about polynomial roots:

• Roots are where the graph intersects the x-axis, representing solutions to the equation.
• Using the Rational Root Theorem, we quickly identify candidates for potential roots.
• After finding these roots, confirm them by synthesizing the division or substitution.
• Each root corresponds to a factor of the polynomial in its factored form, giving \( (x - \text{root}) \).

Understanding the relationships among the roots and factors is crucial in mastering polynomial equations and deepening your algebra proficiency.