Problem 35 - Algebra and Trigonometry

Question

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=2 x^{3}-x^{2}-18 x+9 $$

Step-by-Step Solution

Answer

The factored form is \((2x-1)(x-3)(x+3)\) and the zeros are \(x = \frac{1}{2}, x = 3, x = -3\).

Step 1: Group Terms

First, we will group the terms of the polynomial:\[ P(x) = (2x^3 - x^2) + (-18x + 9) \]

Step 2: Factor by Grouping

Let's factor each group:1. From the first group \(2x^3 - x^2\), factor out \(x^2\): \[ x^2(2x - 1) \]2. From the second group \(-18x + 9\), factor out \(-9\): \[ -9(2x - 1) \]

Step 3: Combine Factored Groups

Both groups have a common factor \(2x - 1\). Factor this out:\[ P(x) = (2x - 1)(x^2 - 9) \]

Step 4: Factor the Quadratic Expression

The expression \(x^2 - 9\) is a difference of squares:\[ x^2 - 9 = (x - 3)(x + 3) \]Thus, the polynomial can be written as:\[ P(x) = (2x - 1)(x - 3)(x + 3) \]

Step 5: Find the Zeros

Set each factor equal to zero to find the zeros of the polynomial:1. \(2x - 1 = 0\) gives \(x = \frac{1}{2}\)2. \(x - 3 = 0\) gives \(x = 3\)3. \(x + 3 = 0\) gives \(x = -3\)

Step 6: Sketch the Graph

Plot the zeros \(x = \frac{1}{2}\), \(x = 3\), and \(x = -3\) on the x-axis. The polynomial is a cubic function, so start from the negative y-direction and make a smooth curve passing through these points and going upward to the positive y-direction as it crosses the x-axis.

Key Concepts

Polynomial Zeros

In the context of polynomial functions, zeros are the x-values where the polynomial evaluates to zero. These points are crucial because they indicate where the graph of the polynomial will intersect or touch the x-axis. Think of them as the solutions to the equation you get when you set the polynomial equal to zero. In our exercise, after factoring the cubic polynomial, we found that its zeros are at

• \(x = \frac{1}{2}\)
• \(x = 3\)
• \(x = -3\)

These zeros are derived from setting each factor of the factored polynomial, \((2x - 1)(x - 3)(x + 3)\), equal to zero and solving for \(x\). Always remember, a polynomial of degree \(n\) will have at most \(n\) real roots, so these zeros completely describe the roots of the cubic function in this particular exercise.

Difference of Squares

The difference of squares is a special case used for factoring certain quadratic expressions. It applies to expressions that can be written in the form \(a^2 - b^2\). Two key points define this pattern:

• The expression is a subtraction \(a^2 - b^2\), not addition.
• Both terms are perfect squares.

For example, in our polynomial, part of the factored form is \(x^2 - 9\). Here, \(x^2\) and \(9\) are both perfect squares, and the expression is in the form of \(a^2 - b^2\). So, we can factor it as:

• \((x - 3)(x + 3)\)

This step is often simple but crucial in breaking down complex polynomials into product forms that make finding zeros and graphing them much easier.

Graphing Cubic Functions

Cubic functions, or third-degree polynomials, have the general form \(ax^3 + bx^2 + cx + d\). They are known for having distinctive shapes due to their polynomial behavior. These functions can have up to three real roots, as shown in our exercise. Characteristics of a cubic function graph include:

• They can intersect the x-axis at up to three points.
• The graph may have up to two turning points, forming peaks and valleys.
• The end behavior typically involves the graph starting in one direction when moving leftward and ending in the opposite direction while moving rightward, forming a sort of 'S' shape.

For our polynomial \(P(x) = 2x^3 - x^2 - 18x + 9\), the graph would intersect the x-axis at its zeros, which are \(\frac{1}{2}, 3, -3\). It starts rising from the negative y-area, descending to pass through these zeros, and ultimately ascends into the positive y-area. Graphing involves not only plotting these zeros but also correctly sketching the curve that passes through them, ensuring the correct direction and turning points, which is key in capturing the essence of cubic behavior.