Problem 39 - Algebra and Trigonometry

Question

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{4}-3 x^{2}-4 $$

Step-by-Step Solution

Answer

The zeros are \( x = -2, 2, i, -i \). The graph crosses the x-axis at \( x = -2 \) and \( x = 2 \).

Step 1: Recognize the Polynomial Structure

The given polynomial is \( P(x) = x^4 - 3x^2 - 4 \). Notice that this resembles a quadratic equation in terms of \( x^2 \). We can treat \( x^4 \) as \( (x^2)^2 \), so the polynomial is a quadratic in terms of \( x^2 \).

Step 2: Substitute to Simplify

Let's make a substitution: let \( u = x^2 \). This changes the polynomial into a simpler quadratic form: \( P(u) = u^2 - 3u - 4 \).

Step 3: Factor the Quadratic Polynomial

We factor the quadratic expression \( u^2 - 3u - 4 \). We look for two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\). Thus, \( u^2 - 3u - 4 = (u - 4)(u + 1) \).

Step 4: Back-Substitute for x

Now substitute \( u \) back with \( x^2 \). This gives us \( (x^2 - 4)(x^2 + 1) \).

Step 5: Solve for Zeros

To find the zeros, we set each factor equal to zero: 1. \( x^2 - 4 = 0 \) implies \( x^2 = 4 \) or \( x = \pm 2 \).2. \( x^2 + 1 = 0 \) implies \( x^2 = -1 \) or \( x = \pm i \). Thus, the zeros of the polynomial are \( x = -2, 2, i, -i \).

Step 6: Sketch the Graph

The polynomial \( P(x) = (x^2 - 4)(x^2 + 1) \) gives real zeros at \( x = -2 \) and \( x = 2 \). The imaginary roots \( x = i, -i \) do not appear on a real graph. The graph of \( P(x) \) will touch or cross the x-axis at \( x = -2 \) and \( x = 2 \), and will be positive as \( x \to \pm \infty \) because the leading term \( x^4 \) is positive. The general shape will be a smooth, W-like curve due to the even power of the polynomial.

Key Concepts

Quadratic Substitution

Quadratic substitution is a powerful technique used to simplify and solve higher-degree polynomials by converting them into a more manageable quadratic form. This method is particularly useful when dealing with polynomials that resemble quadratic equations but are of higher order, such as quartics (degree 4 polynomials).

• Identify the Structure: Look for terms like \(x^4\), which can be seen as \((x^2)^2\). Recognizing this structure is the key to simplifying the problem.
• Substitute Variables: Replace \(x^2\) with a new variable \(u\), which transforms the equation into a quadratic form. For example, the polynomial \(x^4 - 3x^2 - 4\) becomes \(u^2 - 3u - 4\).
• Solve the Quadratic: Once the substitution is made, factor or solve the quadratic to find solutions in terms of \(u\).
• Back-Substitute: Replace \(u\) with \(x^2\) to revert to the original variable, revealing the polynomial's factors in terms of \(x\).

This technique simplifies complex problems, making them more approachable by leveraging familiar quadratic-solving strategies.

Finding Zeros of Polynomials

Finding the zeros of polynomials is an essential task, as they represent the points where the polynomial functions intersect the x-axis in the real number system. To find zeros, set the polynomial equal to zero and solve for x.

• Set Factors to Zero: After factoring the polynomial, equate each factor to zero. For example, with the equation \((x^2 - 4)(x^2 + 1)\), we set \(x^2 - 4 = 0\) and \(x^2 + 1 = 0\).
• Solve Each Equation: Solve these simpler equations for x. From \(x^2 - 4 = 0\), you get \(x = \pm 2\). From \(x^2 + 1 = 0\), you find \(x = \pm i\), using complex numbers when necessary.
• Important Considerations: Real zeros correspond to x-intercepts of the graph, while imaginary/complex zeros do not appear on real coordinate graphs.

Understanding how to find zeros helps in analyzing and sketching polynomial graphs, providing insight into the behavior of polynomial functions.

Graphing Polynomials

Graphing polynomials involves drawing the curve of the polynomial function on a coordinate plane, which visually depicts the polynomial's behavior. It helps to illustrate the zeros and other characteristics like end behavior and turning points.

• Identify Zeros and Intercept: Zeros are the x-values where the polynomial crosses the x-axis (real zeros). For the polynomial \(P(x) = (x^2 - 4)(x^2 + 1)\), the real zeros are \(-2\) and \(2\).
• End Behavior: Consider the leading term, which determines the graph's direction as x approaches positive or negative infinity. Since the leading term is \(x^4\), it suggests that the graph rises to infinity on both sides (positive end behavior).
• Shape and Symmetry: Polynomials of even degree like \(x^4\) tend to have a symmetrical, W-shaped curve. The function's even degree also informs that it will rise on both ends.

When graphing, start by plotting the real zeros and sketching the general shape. Remember that complex zeros, like in this case, do not affect the real graph but are still critical for understanding the full scope of the polynomial.