Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Simplify \(f(x)\), and sketch the graph of \(f\). $$ f(x)=\frac{x^{3}-2 x^{2}-4 x+8}{x-2} $$

Step-by-Step Solution

Answer

The simplified function is \( x^2 - 4 \), a parabola with vertex at \((0, -4)\), opening upwards, and intercepts at \( x = 2 \) and \( x = -2 \).

Step 1: Identify Polynomial Division

The function given is \( f(x) = \frac{x^{3}-2x^{2}-4x+8}{x-2} \). This requires polynomial division since we are dividing a cubic polynomial by a linear polynomial.

Step 2: Perform Synthetic Division

Perform synthetic division to divide the cubic polynomial \( x^3 - 2x^2 - 4x + 8 \) by \( x-2 \). Set up the synthetic division:1. Use 2 in the synthetic division setup (since \( x - 2 = 0 \) results in \( x = 2 \)).2. Coefficients of the polynomial are: \( 1, -2, -4, 8 \).3. The synthetic division steps: - Bring down the leading coefficient (1). - Multiply 2 (from \( x-2 \)) by 1 to get 2, add this to the next coefficient (-2) to get 0. - Multiply 2 by 0 to get 0, add this to the next coefficient (-4) to get -4. - Multiply 2 by -4 to get -8, add this to the last coefficient (8) to get 0.4. The result of synthetic division is \( x^2 - 4 \) with a remainder of 0.

Step 3: Write Simplified Expression

Since the remainder is 0 from synthetic division, the polynomial \( x^3 - 2x^2 - 4x + 8 \) is perfectly divisible by \( x - 2 \). Thus, the simplified expression of \( f(x) \) is \( f(x) = x^2 - 4 \).

Step 4: Recognize Structure of Simplified Function

Notice that \( x^2 - 4 \) can be further simplified using the difference of squares: \[ f(x) = (x-2)(x+2) \]. This shows that the graph will be a parabola, with x-intercepts at \( x = 2 \) and \( x = -2 \).

Step 5: Sketch the Graph

The function \( f(x) = x^2 - 4 \) is a parabola opening upwards. - The vertex is at the origin: \( (0, -4) \).- The x-intercepts are at \( x = 2 \) and \( x = -2 \).- Since the coefficient of \( x^2 \) is positive, the parabola opens upwards.Sketch this by marking the vertex, plotting the intercepts, and drawing a symmetric parabola through these points.

Key Concepts

Synthetic Division

Synthetic Division is a simplified form of polynomial division, used primarily when dividing a polynomial by a first-degree binomial such as \(x - c\). This method is favored for its straightforward application and efficiency.
To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial you're dividing.
• Identify the value of \(c\) by setting \(x - c = 0\), which makes \(x = c\).
• Using \(c\), apply it to each coefficient in a process that involves multiplying and adding successively. This process continues until you finish with a remainder.

This method notably simplifies our division, especially with higher-degree polynomials. In our example, the cubic polynomial \(x^3 - 2x^2 - 4x + 8\) divides cleanly by \(x - 2\), yielding a quotient of \(x^2 - 4\) without any remainder.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, taking the general form \(ax^3 + bx^2 + cx + d\). These polynomials have a leading coefficient that determines their highest degree term, \(x^3\).
Cubic polynomials can have up to three real roots and can display complex behavior, potentially crossing the x-axis three times. In the given exercise, the polynomial is \(x^3 - 2x^2 - 4x + 8\), and we were tasked to simplify it using synthetic division.
By dividing this cubic polynomial by \(x-2\), we test if \(x=2\) is a root. The remainder of zero from the division confirms that it indeed is, demonstrating that the polynomial is perfectly divisible, simplifying our analysis.

Difference of Squares

Once we simplify \(x^3 - 2x^2 - 4x + 8\) to \(x^2 - 4\) using synthetic division, we can further break down the expression using the Difference of Squares formula.
The formula states that \(a^2 - b^2 = (a - b)(a + b)\). This is applicable when the polynomial is a difference between two perfect squares.
In our example, \(x^2 - 4\) is a difference of squares since it can be expressed as \(x^2 - 2^2\). Consequently, using the formula yields \((x - 2)(x + 2)\).

• This factorization reveals the roots of the quadratic, which are also the x-intercepts of the graph: \(x = 2\) and \(x = -2\).
• Understanding this principle assists in both algebraic simplification and graphical representation.

Parabola Graph Sketching

Graph sketching allows us to visually interpret equations and understand their behavior.
When a polynomial simplifies to \(x^2 - 4\) - or \((x - 2)(x + 2)\) - it describes a parabola opening upwards because the \(x^2\) term coefficient is positive. The graph of \(x^2 - 4\) has specific characteristics:

• The vertex of this parabola is at the point \((0, -4)\), where it intersects the y-axis.
• The x-intercepts, where the parabola crosses the x-axis, are at \(x = 2\) and \(x = -2\).
• Since the coefficient of \(x^2\) is 1 (a positive number), this tells us that our parabola will 'open' upwards, creating a "U" shape.

Understanding these points and the parabola's symmetry aids in drawing it accurately. Once the x-intercepts and vertex are marked, connecting them with a smooth curve completes the sketch, illustrating the full behavior of the function.