Problem 33 - Precalculus Mathematics for Calculus

Question

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{5}+6 x^{3}+9 x$$

Step-by-Step Solution

Answer

Zeros are 0, \(i\sqrt{3}\), and \(-i\sqrt{3}\) with multiplicities 1, 2, and 2.

Step 1: Identify Common Factors

First, we look for any common factors in all the terms of the polynomial \(P(x) = x^5 + 6x^3 + 9x\). We notice that each term contains an \(x\). Thus, we can factor out \(x\) as a common factor: \[ P(x) = x(x^4 + 6x^2 + 9) \]

Step 2: Factor Further

Next, we factor the quadratic \(x^4 + 6x^2 + 9\). Notice that the expression can be written in terms of \((x^2)^2\), so we try factoring as a quadratic: \[ x^4 + 6x^2 + 9 = (x^2 + 3)(x^2 + 3) = (x^2 + 3)^2 \] Therefore, \(P(x) = x(x^2 + 3)^2\).

Step 3: Find the Zeros

To find the zeros of \(P(x) = x(x^2 + 3)^2\), set each factor equal to zero: \[ x = 0 \] \[ (x^2 + 3)^2 = 0 \] Solving \(x^2 + 3 = 0\), we get: \[ x^2 = -3 \] \[ x = i\sqrt{3}, \, -i\sqrt{3} \] Thus, the zeros are \(x = 0\), \(x = i\sqrt{3}\), and \(x = -i\sqrt{3}\).

Step 4: Determine Multiplicity of Zeros

The multiplicity of each zero is determined by the exponent on the factor: - The zero \(x = 0\) corresponds to \(x^1\) and thus has multiplicity 1. - The zeros \(x = i\sqrt{3}\) and \(x = -i\sqrt{3}\) each come from \((x^2 + 3)^2\), both with multiplicity 2.

Key Concepts

zeros of a polynomial

Zeros of a polynomial are the points where the polynomial equals zero. In other words, if you substitute a zero into the polynomial function, the output will be zero. It's like finding the roots of an equation. To determine the zeros of a polynomial, we set the polynomial equal to zero and solve for the variable.

Consider the polynomial \( P(x) = x(x^2 + 3)^2 \):

• The first factor, \( x \), gives us a zero at \( x = 0 \).
• The second factor, \( (x^2 + 3)^2 \), involves solving \( x^2 + 3 = 0 \).

Solving \( x^2 + 3 = 0 \) necessitates moving 3 to the other side, resulting in \( x^2 = -3 \). Since squares of real numbers can’t be negative, we enter the realm of complex numbers for these solutions. Thus, we find:

• \( x = \pm i\sqrt{3} \).

In this case, the polynomial has both real and imaginary zeros. Understanding this helps identify where the graph of the polynomial will intersect the x-axis (for real zeros) or exist in the complex plane.

multiplicity of zeros

Multiplicity refers to the number of times a particular zero appears in the factored form of a polynomial. Each factor of the polynomial contributes not only a zero but specifies how that zero affects the graph.

For the polynomial \( P(x) = x(x^2 + 3)^2 \):

• The zero \( x = 0 \) is associated with the factor \( x \) raised to the power of 1, giving it a multiplicity of 1. It means the graph crosses the x-axis at this point.
• The zeros \( x = i\sqrt{3} \) and \( x = -i\sqrt{3} \) are associated with the factor \( (x^2 + 3)^2 \), each having a multiplicity of 2. These zeros suggest the polynomial has complex roots, where each is counted twice.

The multiplicity of a zero influences the behavior of the graph at the x-axis.
- If the multiplicity is odd, the graph crosses the x-axis at the zero.
- If it's even, the graph just touches the axis and bounces back.
Understanding multiplicity is crucial when predicting graph behavior near a zero.

complex numbers in polynomials

Complex numbers frequently appear when solving polynomials that include negative square roots, as they offer solutions where real numbers do not suffice. They consist of a real part and an imaginary part, usually represented as \( a + bi \), where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

In the polynomial \( P(x) = x(x^2 + 3)^2 \), solving for zeros in the factor \( x^2 + 3 = 0 \) leads us to complex numbers:

• \( x = \pm i\sqrt{3} \)

These are imaginary numbers, specifically because they can’t be visualized on the traditional x-y axis graph.

Complex zeros often come in conjugate pairs. For example, if \( x = i\sqrt{3} \) is a zero, then \( x = -i\sqrt{3} \) will also be a zero.
Understanding complex numbers is crucial for a complete analysis of polynomial functions, especially those which do not have all real solutions. They expand our ability to find all possible roots, ensuring we capture every potential solution the polynomial provides.