Problem 29 - Algebra and Trigonometry

Question

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{3}+x^{2}-x-1 $$

Step-by-Step Solution

Answer

Factored form: \((x + 1)^2(x - 1)\); Zeros: \(x = -1\) (multiplicity 2), \(x = 1\).

Step 1: Group terms

First, we will group the given polynomial terms to make it easier for factoring. We begin with grouping the terms as follows: \((x^3 + x^2) + (-x - 1)\).

Step 2: Factor by grouping

In the expression \((x^3 + x^2) + (-x - 1)\), factor out the common factors in each group. \(x^2(x + 1) - 1(x + 1)\).

Step 3: Factor out the common binomial factor

Notice that \((x + 1)\) is a common factor in both groups. Factor \((x + 1)\) out: \((x + 1)(x^2 - 1)\).

Step 4: Factor the quadratic

The quadratic \((x^2 - 1)\) can be factored as a difference of squares: \((x - 1)(x + 1)\).

Step 5: Write the fully factored form

Using the factored parts, the fully factored form of the polynomial is \((x + 1)^2(x - 1)\).

Step 6: Find the zeros

Set the factored form equal to zero: \((x + 1)^2(x - 1) = 0\). The solutions are \(x = -1\) and \(x = 1\). The zero \(x = -1\) has a multiplicity of 2.

Step 7: Sketch the graph

To sketch the graph, note the zeros and their multiplicities: \(x = -1\) (double root) and \(x = 1\) (single root). The graph touches the x-axis at \(x = -1\) (due to the double root) and crosses the x-axis at \(x = 1\). The end behavior will rise to the right and fall to the left since it is a cubic polynomial with a positive leading coefficient.

Key Concepts

Cubic Polynomial

A cubic polynomial is a type of polynomial equation that involves a variable raised to the third power. In this case, our cubic polynomial is given by \( P(x) = x^3 + x^2 - x - 1 \). Most often, cubic polynomials have the general form \( ax^3 + bx^2 + cx + d \), where the highest degree is 3. This degree tells you the number of roots or solutions the polynomial can have, including complex roots. It's important to remember that cubic polynomials can have up to three real roots. The leading coefficient (here, 1) of a cubic polynomial affects the end behavior of its graph. For the polynomial \( P(x) \), with a positive leading coefficient, the graph's tail end will rise on the right and fall on the left. Understanding the structure and behavior of cubic polynomials is crucial during the process of graphing and finding their zeros.

Difference of Squares

The difference of squares is a factoring technique used when you have two perfect squares separated by a subtraction sign. The formula for factoring a difference of squares is \( a^2 - b^2 = (a - b)(a + b) \). In our polynomial problem, we use this technique on the quadratic expression \( x^2 - 1 \). Here, both \( x^2 \) and 1 are perfect squares because \( x^2 = (x)^2 \) and \( 1 = (1)^2 \). By applying the difference of squares factoring method, we can rewrite \( x^2 - 1 \) as \( (x - 1)(x + 1) \). This step is critical because it breaks down the expression into simpler factors that are easier to manage in the process of finding the zeros of the polynomial.

Multiplicity of Roots

Multiplicity of roots refers to how many times a particular root appears in the solution of a polynomial equation. When factoring the cubic polynomial \( P(x) \), it becomes \( (x + 1)^2(x - 1) \). Here, \( x = -1 \) is a repeated root, which means it has a multiplicity of 2, while \( x = 1 \) is a single root. Multiplicity affects the polynomial's graph: a root with a multiplicity of 2 or more results in the graph merely touching the x-axis at that root, rather than crossing it. In this example, the graph touches the x-axis at \( x = -1 \), indicating a double root (multiplicity of 2), and crosses the x-axis at \( x = 1 \), indicating a root with multiplicity of 1. Understanding root multiplicity is essential for accurately sketching the graph of the polynomial function.

Zeros of a Polynomial

Zeros of a polynomial are the x-values at which the polynomial equals zero. Finding these zeros is a key step in factoring and graphing polynomials. For the polynomial \( P(x) = x^3 + x^2 - x - 1 \), we fully factored it as \( (x + 1)^2(x - 1) \). Setting each factor equal to zero gives the zeros, namely \( x = -1 \) and \( x = 1 \). Zeros indicate where the graph of the polynomial will intersect or touch the x-axis. For \( x = -1 \), the graph will only touch the x-axis because it is a zero with multiplicity 2. Alternatively, for \( x = 1 \), the graph will cross the x-axis, as it is a zero with multiplicity 1. Recognizing these zeros is crucial not only for drawing the graph but also for understanding the behavior and the roots of the polynomial function.