Problem 30 - Precalculus Mathematics for Calculus

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 polynomial is factored as \(x(-2x + 1)(x + 1)\) and its zeros are \(x = 0, \frac{1}{2}, -1\).

Step 1: Factor the Polynomial

To factor the polynomial \(P(x) = -2x^3 - x^2 + x\), first look for a common factor in all terms. Notice that each term contains \(x\). Factor out \(x\) to get: \[ P(x) = x(-2x^2 - x + 1). \]Next, focus on factoring the quadratic expression \(-2x^2 - x + 1\). Look for two numbers that multiply to \(-2 \times 1 = -2\) and add up to \(-1\). These numbers are \(-2\) and \(1\). So, rewrite the quadratic expression as: \[ -2x^2 - 2x + x + 1. \]Factor by grouping: \[ -2x(x + 1) + 1(x + 1). \]This gives: \[ (-2x + 1)(x + 1). \]Therefore, the fully factored form of \(P(x)\) is:\[ P(x) = x(-2x + 1)(x + 1). \]

Step 2: Find the Zeros of the Polynomial

To find the zeros of the polynomial, set each factor equal to zero and solve for \(x\):1. \(x = 0\)2. \(-2x + 1 = 0 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}\)3. \(x + 1 = 0 \Rightarrow x = -1\)The zeros of the polynomial are \(x = 0, \frac{1}{2}, -1\).

Step 3: Sketch the Graph

To sketch the graph of the polynomial, note that it is a cubic polynomial and observe the factored form \(P(x) = x(-2x + 1)(x + 1)\). The zeros are \(x = 0, \frac{1}{2}, -1\). The leading coefficient is negative (\(-2\)), so the graph will fall to the right and rise to the left. Plot the zeros on the x-axis, and draw a smooth curve passing through these points, starting from below the x-axis, rising through \(x = -1\), and returning below the x-axis after \(x = \frac{1}{2}\).

Key Concepts

Finding Zeros of Polynomials

When you're tasked with finding the zeros of a polynomial, you're essentially trying to discover the x-values at which the polynomial equals zero. For the polynomial \( P(x) = -2x^3 - x^2 + x \), we first factor it as \( P(x) = x(-2x + 1)(x + 1) \). To find the zeros, we solve each factor set to zero:

• \( x = 0 \)
• \(-2x + 1 = 0 \Rightarrow x = \frac{1}{2}\)
• \( x + 1 = 0 \Rightarrow x = -1 \)

These solutions give us the zeros of the polynomial, which are the points where the graph crosses or touches the x-axis. Understanding the zeros helps in sketching the graph and analyzing the behavior of the polynomial.

Quadratic Expression Factoring

Factoring a quadratic expression is key to simplifying polynomials and finding their zeros. In our exercise, the quadratic part is \(-2x^2 - x + 1\). The goal is to break it down into simpler factors. To factor, look for numbers that multiply to the product of the leading coefficient and constant term. Here,

• Product is \(-2 \times 1 = -2\)
• Numbers are \(-2\) and \(1\)

These numbers can rewrite the middle term: \(-2x^2 - 2x + x + 1\). Factoring by grouping, we extract common elements: \(-2x(x + 1) + 1(x + 1)\). Bringing it all together, we have: \((-2x + 1)(x + 1)\). This process is crucial, as it simplifies solving polynomials by creating smaller, manageable expressions.

Polynomial Graph Sketching

Graph sketching of polynomials provides a visual representation of how they behave. For \( P(x) = x(-2x + 1)(x + 1) \), you begin by identifying key features:

• The zeros: \( x = 0, -1, \frac{1}{2} \)
• The leading coefficient (-2), which is negative, indicating the direction of the tails

Cubic polynomials tend to start from one end and extend to the other.
Here, because the leading coefficient is negative, the polynomial falls to the right and rises to the left. Sketching involves:

• Plotting the zeros on the x-axis
• Checking the polynomial value between zeros
• Drawing a smooth curve that passes through these points in alignment with the behavior highlighted by the negative leading coefficient

This visual can be invaluable for understanding the overall trend and specific characteristics of the polynomial function.