Problem 30 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The zeros are \( x = 0, -1, \frac{1}{2} \). Graph descends from left, turning at zeros.

Step 1: Factor Out Common Terms

Examine the polynomial \( P(x) = -2x^3 - x^2 + x \). Notice that each term contains an \( x \), which can be factored out. Thus, the polynomial becomes:\[ P(x) = x(-2x^2 - x + 1) \].

Step 2: Factor the Quadratic Expression

Now, focus on the quadratic expression \(-2x^2 - x + 1\). We need to factor this expression. Look for two numbers that multiply to \(-2 \times 1 = -2\) and add up to \(-1\). These numbers are \(-2\) and \(1\). Split the middle term using these numbers:\[ -2x^2 - 2x + x + 1 \].Group the terms:\[ (-2x^2 - 2x) + (x + 1) \].Factor out the greatest common factor from each group:\[ -2x(x + 1) + 1(x + 1) \].Finally, factor by grouping:\[ (x + 1)(-2x + 1) \].

Step 3: Write the Fully Factored Form

Combine the factored linear term from Step 1 with the factored quadratic expression from Step 2 to write the fully factored form of the polynomial:\[ P(x) = x(x + 1)(-2x + 1) \].

Step 4: Find the Zeros of the Polynomial

The zeros of the polynomial occur where \( P(x) = 0 \). Set each factor equal to zero and solve for \( x \):1. \( x = 0 \).2. \( x + 1 = 0 \) implies \( x = -1 \).3. \(-2x + 1 = 0 \) implies \( -2x = -1 \), thus \( x = \frac{1}{2} \).So, the zeros are \( x = 0, -1, \frac{1}{2} \).

Step 5: Sketch the Graph

On the sketch, plot the zeros \( x = 0, -1, \frac{1}{2} \) on the x-axis. Since the leading coefficient of the original polynomial is negative, the graph should start from the negative region in the left and end in the negative region on the right (downward opening proportionally to the order of the polynomial). Pass through the zeros with the general cubic shape characteristic of the function \(-2x^3\).

Key Concepts

Polynomial Zeros

Polynomial zeros are the values of the variable that make the polynomial equal to zero. These zeros are important because they represent the points where the graph of the polynomial will intersect the x-axis. To find the zeros, we solve the equation set by the fully factored form of the polynomial. In this case, the polynomial is fully factored as: \[ P(x) = x(x + 1)(-2x + 1) \] To find the zeros, set each factor equal to zero and solve for \( x \):

• \( x = 0 \): This is straightforward as the term \( x \) already equals zero.
• \( x + 1 = 0 \): Solving this gives \( x = -1 \).
• \( -2x + 1 = 0 \): Solving this gives \( x = \frac{1}{2} \).

Thus, the zeros of the polynomial are \( x = 0, -1, \frac{1}{2} \). These zeros are fundamental for plotting the graph, as they define where the curve will touch the x-axis.

Graph Sketching

Graph sketching involves drawing a rough diagram of the polynomial based on its zeros and behavior. For the polynomial \( P(x) = -2x^3 - x^2 + x \), after factoring and finding the zeros, we made a clearer path for sketching its graph.When sketching a graph:

• Plot the zeros on the x-axis. For \( P(x) \), these are \( x = 0, -1, \frac{1}{2} \).
• Consider the end behavior of the polynomial. Since the leading coefficient of the original polynomial is \(-2\), which is negative and not even, it shows that as \( x \) moves to either extreme negative or positive, the graph will descend downwards.

The general shape for a cubic equation like this one is a curve starting from the negative on the left, rising as it passes through the zeros, and ending negatively on the right. Sketching involves ensuring the smooth curve reflects these movements, passing through all zeros clearly.

Quadratic Expression Factoring

Factoring quadratic expressions is an essential skill in algebra that helps in simplifying equations or finding solutions like zeros. Looking at our problem, we had the quadratic expression \[ -2x^2 - x + 1 \]. Factor this expression using the steps below:

• Find two numbers that multiply to \(-2 \cdot 1 = -2\) and add to \(-1\). These numbers are \(-2\) and \(1\).
• Split the middle term \( -x \) using these numbers to form \(-2x^2 - 2x + x + 1\).
• Group and factor each resulting pair. This step involves factoring common terms from groups: \[ (-2x^2 - 2x) + (x + 1) \].
• Factor each group individually to get: \[ -2x(x + 1) + 1(x + 1) \].
• Notice \((x + 1)\) being common, and factor by grouping to obtain \((x + 1)(-2x + 1)\).

Factoring this way simplifies solving the polynomial as it reveals zeros and aids greatly in graph sketching and further mathematical operations.