Problem 26 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ Q(x)=x^{4}+10 x^{2}+25 $$

Step-by-Step Solution

Answer

Zeros are \( x = \pm i\sqrt{5} \) with multiplicity 2 each.

Step 1: Recognize the Polynomial Form

The polynomial given is \( Q(x) = x^4 + 10x^2 + 25 \). Notice that it is a quadratic in terms of \( x^2 \). This is a quadratic form because \( x^4 = (x^2)^2 \).

Step 2: Substitute to Simplify

Let \( u = x^2 \). This substitution transforms the polynomial into \( Q(u) = u^2 + 10u + 25 \).

Step 3: Factor the Quadratic

The expression \( u^2 + 10u + 25 \) is a perfect square trinomial. It can be factored as \( (u + 5)^2 \).

Step 4: Substitute Back

Replace \( u \) with \( x^2 \) in the factored form. Therefore, \( Q(x) = (x^2 + 5)^2 \).

Step 5: Find the Zeros of the Polynomial

Set \( Q(x) = (x^2 + 5)^2 = 0 \). This implies \( x^2 + 5 = 0 \). Solving for \( x \), we have \( x^2 = -5 \). This gives the solution \( x = \pm i\sqrt{5} \).

Step 6: Determine the Multiplicity of Each Zero

Since the term \( (x^2 + 5) \) is squared, each zero \( x = i\sqrt{5} \) and \( x = -i\sqrt{5} \) has a multiplicity of 2.

Key Concepts

Quadratic Form

When faced with polynomials such as \( Q(x) = x^4 + 10x^2 + 25 \), it's important to identify the pattern it follows. In this case, the polynomial is in a **quadratic form** because it can be expressed in terms of another variable squared. This occurs because the highest power variable, \( x^4 \), can be rewritten as \((x^2)^2\). By recognizing this structure, it becomes easier to handle the polynomial by treating it like a standard quadratic equation.

To simplify, substitute \( u = x^2 \), which transforms the original polynomial into \( Q(u) = u^2 + 10u + 25 \). Identifying a polynomial as a quadratic form is a critical step because it allows the polynomial to be factored using methods applicable to quadratic equations.

Zeros of Polynomial

Finding the zeros of a polynomial involves determining the values of \( x \) that make the polynomial equal to zero. Zeros are important because they indicate where the graph of the polynomial intersects the x-axis. For the polynomial \( Q(x) = (x^2 + 5)^2 \), set it equal to zero to find its zeros: \((x^2 + 5)^2 = 0\). This implies \( x^2 + 5 = 0 \).

Solving \( x^2 + 5 = 0 \) gives \( x^2 = -5 \). Since no real number squared results in a negative, we turn to complex numbers: \( x = \pm i\sqrt{5} \). These complex numbers are the zeros of the polynomial. The process of finding polynomial zeros is about solving for the roots of the equation, often requiring factoring, graphing, or using algebraic properties.

Multiplicity of Zeros

The **multiplicity** of a zero in a polynomial refers to the number of times a particular zero appears as a solution. It is determined by the exponent of the factor corresponding to the zero. In \( Q(x) = (x^2 + 5)^2 \), the factor \((x^2 + 5)\) is squared, indicating that each of its zeros occurs twice, hence each has a multiplicity of 2.

Understanding multiplicity is key for analyzing the behavior of polynomial graphs. A zero with an odd multiplicity will cause the graph to cross the x-axis, while an even multiplicity means the graph just touches the x-axis and turns around. In this example, since \( x = i\sqrt{5} \) and \( x = -i\sqrt{5} \) both have a multiplicity of 2, these points would result in the graph only touching the x-axis at these roots if they were real numbers, demonstrating how multiplicity informs graphical interpretations.