Problem 27 - Algebra for College Students

Question

For Problems \(23-34\), graph each polynomial function by first factoring the given polynomial. You may need to use some factoring techniques from Chapter 3 as well as the rational root theorem and the factor theorem. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

Step-by-Step Solution

Answer

The polynomial function is factored as \(f(x) = (x - 1)(x + 1)(x + 2)\).

Step 1: Identify and List Possible Rational Roots

Using the rational root theorem, list the potential rational roots of the polynomial \(f(x) = x^3 + 2x^2 - x - 2\). These are factors of the constant term (-2) divided by the leading coefficient (1). Thus, the possible rational roots are \(\pm 1, \pm 2\).

Step 2: Test Possible Rational Roots

Substitute the possible rational roots into the polynomial to find an actual root. Begin testing with \(x = 1\): \[ f(1) = 1^3 + 2(1)^2 - 1 - 2 = 1 + 2 - 1 - 2 = 0 \]So, \(x = 1\) is a root.

Step 3: Perform Synthetic Division

Use synthetic division to divide the polynomial \(f(x) = x^3 + 2x^2 - x - 2\) by \(x - 1\). The division results in a quotient of \(x^2 + 3x + 2\).

Step 4: Factor the Quotient

Factor the quadratic function \(x^2 + 3x + 2\). The factors are \((x + 1)(x + 2)\).

Step 5: Write the Factored Form of the Polynomial

Combine the root from Step 2 and the factors from Step 4 to get the completely factored form of \(f(x)\):\[ f(x) = (x - 1)(x + 1)(x + 2) \]

Step 6: Graph the Polynomial Function

The roots of the polynomial are \(x = 1\), \(x = -1\), and \(x = -2\). Plot these roots on the x-axis. Sketch the polynomial graph, noting that as a cubic function with a positive leading coefficient, the graph starts from negative infinity (left) and ends at positive infinity (right). The graph crosses the x-axis at each root.

Key Concepts

Factoring Techniques

Factoring polynomial functions is essential when you want to simplify them or find their roots. Polynomials can often be factored into simpler expressions using several techniques. The goal is to rewrite the polynomial as a product of simpler polynomials.
Some basic techniques include:

• Factoring by grouping: This involves rearranging and grouping terms to find common factors.
• Factoring quadratic polynomials: For quadratics in the form of ax² + bx + c, finding two numbers that multiply to ac and add to b helps in splitting the middle term.
• Using special identities: Such as the difference of squares, perfect square trinomials, or the sum/difference of cubes.

In this specific exercise, we applied factoring techniques to deduce that the quadratic quotient from synthetic division was factorable into (x + 1)(x + 2). This process is critical for solving and graphing polynomial functions.

Rational Root Theorem

The Rational Root Theorem is a handy tool in determining the potential rational roots of a polynomial function. According to this theorem, any rational solution, expressed as a fraction \( \frac{p}{q} \), will have its numerator \( p \) as a factor of the constant term and its denominator \( q \) as a factor of the leading coefficient.
In our exercise, we identified that possible rational roots for \( f(x) = x^3 + 2x^2 - x - 2 \) include \( \pm 1, \pm 2 \). We confirm these potential roots by testing each value in the function until we find the root that makes the polynomial equal zero. Once a root is found, we know that \( x - \) that root \( \,\) is a factor of the polynomial.
Utilizing the Rational Root Theorem is often the first step in factoring polynomial functions.

Synthetic Division

Synthetic division is a simplified way to divide polynomials when you know one of the roots or when using potential roots from the Rational Root Theorem. It is faster and more efficient than long division.
In synthetic division:

• Write down the coefficients of the polynomial.
• Bring down the leading coefficient.
• Multiply the current value by the root used and add it to the next coefficient.
• Continue this process across all coefficients to find the quotient and remainder.

When you perform synthetic division with an actual root, like in our example with root \( x=1 \), the remainder should be zero, confirming it's a true root. This leaves you with the remaining quotient polynomial, which can then be further factored if it's simpler, such as a quadratic.

Graphing Polynomial Functions

Graphing polynomial functions provides a visual representation of the roots and the behavior of the polynomial. Each root corresponds to an x-intercept of the graph. The degree of the polynomial gives insight into the number of roots and the general shape of the graph.
For example, a cubic polynomial (degree 3) will generally have a classic 'S' shape, with three roots crossing the x-axis. It can have:

• Inflection points, where the curvature changes direction.
• End behavior determined by the leading coefficient and degree.

Our function \( f(x) = x^3 + 2x^2 - x - 2 \) was factored as \( (x - 1)(x + 1)(x + 2) \). This shows that the roots are \( x = 1, -1, -2 \). When graphing, the function crosses the x-axis at these points. The cubic nature and positive leading coefficient tell us the graph rises from left to right.