Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The factored form is \(x(x + 1)(-2x + 1)\); zeros are \(x = 0, -1, 0.5\). Sketch shows graph crossing at these points.

Step 1: Factor Out the Greatest Common Factor (GCF)

Start by identifying and factoring out the greatest common factor from the polynomial. Looking at the polynomial \(-2x^3 - x^2 + x\), each term has a common factor of \(x\). Factoring out \(x\), we get \(x(-2x^2 - x + 1)\).

Step 2: Factor the Quadratic Expression

Consider the quadratic \(-2x^2 - x + 1\) inside the parentheses. To factor this quadratic, look for two numbers that multiply to \(-2 \cdot 1 = -2\) and add to \(-1\). These numbers are \(-2\) and \(1\). Rewrite the quadratic as \(-2x^2 - 2x + x + 1\), then factor by grouping.Group the terms: \(-2x^2 - 2x\) and \(x + 1\).Factor each group: \(-2x(x + 1) + 1(x + 1)\).Factor out the common binomial \((x + 1)\): \((x + 1)(-2x + 1)\).

Step 3: Write the Complete Factored Form

Now, include the factored GCF \(x\) from Step 1. The completely factored form of the polynomial is:\[P(x) = x(x + 1)(-2x + 1)\]

Step 4: Find the Zeros of the Polynomial

The zeros of the polynomial are found by setting each factor equal to zero:\(x = 0\), \(x + 1 = 0 \Rightarrow x = -1\), \(-2x + 1 = 0 \Rightarrow x = \frac{1}{2}\).Thus, the zeros of \(P(x)\) are \(x = 0, -1, \frac{1}{2}\).

Step 5: Sketch the Graph

Using the zeros \(x = 0, -1, 0.5\), sketch a graph of the polynomial. Note the behavior around these zeros:- At \(x = 0\), the graph crosses the x-axis.- At \(x = -1\) and \(x = 0.5\), the graph also crosses the x-axis since it is a cubic polynomial with a negative leading coefficient, which creates an end behavior of the graph going from positive infinity to negative infinity.

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. These are also known as the roots or solutions of the polynomial equation. For a polynomial expressed in the factored form \(P(x) = (x-a)(x-b)(x-c)\), the zeros are \(x = a, b, c\).
To find the zeros, we set each factor of the polynomial equal to zero and solve for the variable. This is a key step because the zeros of a polynomial indicate where the graph of the polynomial crosses or touches the x-axis.
In this exercise, after factoring, we determined that the zeros are \(x = 0\), \(x = -1\), and \(x = \frac{1}{2}\). Determining these points is crucial, especially when you need to sketch the graph and understand how the polynomial behaves across different intervals.

Graphing Polynomials

Graphing polynomials involves plotting the curve of the polynomial function on a coordinate plane. Each polynomial has its own unique shape, and this can be largely determined by its degree, the leading coefficient, and its zeros.
Here’s a simple approach to graphing a polynomial:

• Identify the zeros of the polynomial. In this case, they are \(x = 0, -1, \frac{1}{2}\).
• Determine the end behavior of the polynomial, which for cubic polynomials is influenced by the leading coefficient. A negative leading coefficient means the graph starts in positive infinity and ends in negative infinity.
• Plot additional points between and beyond the zeros to form a clearer picture of the graph.

This helps in visualizing how the polynomial changes and where it will intersect with the x-axis, resulting in a more accurate depiction of its behavior.

Cubic Polynomial

A cubic polynomial is a polynomial of degree 3, meaning the highest exponent of the variable is three. An example of a cubic polynomial is \(P(x) = -2x^3-x^2+x\).
Some important characteristics of cubic polynomials include:

• They can have up to three zeros.
• Their graphs can have up to two turning points.
• The end behavior is determined by the leading coefficient (negative leading coefficients result in a significant decrease from left to right).

The properties of cubic polynomials make them quite interesting, as they often represent real-world scenarios where there is change, such as growth and decay. Understanding how to factor and graph cubic polynomials builds a solid foundation for analyzing more complex mathematics.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. When factoring polynomials, identifying the GCF can simplify the expression by removing common factors from each term.
For example, in the polynomial \(P(x) = -2x^3-x^2+x\), the GCF of all terms is \(x\), since each term is divisible by \(x\). Factoring out the GCF involves creating a simpler expression: \(x(-2x^2-x+1)\).
This step is crucial because it simplifies the polynomial and makes it easier to factor further. By understanding the GCF, you make the process of factoring more efficient and less prone to errors, which is key when solving polynomial equations.