Problem 13 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The zeros are \(-5i\) and \(5i\), each with multiplicity 1.

Step 1: Understand the Polynomial

We are given the polynomial \( P(x) = x^2 + 25 \). Note that this is a quadratic polynomial and needs to be factored to find its zeros.

Step 2: Identify the Form and Attempt Factoring

The polynomial \( x^2 + 25 \) is in the form \( x^2 + a^2 \), where \( a = 5 \). Note that this form cannot be factored using real numbers. Instead, we use complex numbers.

Step 3: Use Complex Number Factorization

The expression \( x^2 + 25 \) can be factored using complex numbers as \( (x + 5i)(x - 5i) \). This is done by recognizing it fits the sum of squares, which factors similarly to the difference of squares, but with imaginary units \( i \).

Step 4: Find the Zeros of the Polynomial

Set each factor equal to zero to find the zeros. For \( (x + 5i) \), set \( x + 5i = 0 \) which gives \( x = -5i \). For \( (x - 5i) \), set \( x - 5i = 0 \) which gives \( x = 5i \).

Step 5: Determine the Multiplicity of Each Zero

Both roots \( x = -5i \) and \( x = 5i \) are derived from linear factors and appear only once. Therefore, they each have a multiplicity of 1.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a polynomial of the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). In our given polynomial \(P(x) = x^2 + 25\), the structure is simplified to \(x^2 + 0x + 25\). This indicates that it is a simple quadratic without the linear term.
Understanding how to identify a quadratic polynomial is the first step in tackling polynomial factorization. Notably, any expression that fits the \(x^2\) form without an \(x\) or constant term being zero is quadratic.

• The highest power of \(x\) is 2, making it a quadratic.
• The quadratic component \(x^2\) dictates the polynomial's fundamental properties.

Identifying the type of polynomial you are dealing with is crucial before moving on to factorization.

Complex Numbers

Complex numbers expand the set of solutions we can find for polynomial equations, especially when dealing with quadratics that cannot be factored using real numbers. A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit \(i^2 = -1\).
In the context of our polynomial \(x^2 + 25\), the roots involve the imaginary unit \(i\). This is because the expression can be seen as a sum of squares, specifically \(x^2 + (5i)^2\).

• Complex numbers help solve equations with no real roots.
• They reveal solutions in real-world applications like physics and engineering.

Using complex numbers, we factor the expression as \((x + 5i)(x - 5i)\), demonstrating their essential role in algebra.

Zeros of a Polynomial

Zeros of a polynomial are values of \(x\) that make the polynomial equal to zero. For quadratic polynomials like \(P(x) = x^2 + 25\), identifying the zeros involves solving the equation \(P(x) = 0\).
For our polynomial, the factorization \((x + 5i)(x - 5i) = 0\) gives us two zeros: \(x = -5i\) and \(x = 5i\). These zeros are complex, highlighting the importance of understanding and applying complex number arithmetic.

• Zeros represent the x-values where the polynomial crosses or just touches the x-axis.
• Even if the graph does not visually intersect the axis, as with imaginary or complex zeros, they exist in an extended dimension.

Finding zeros is key to understanding the characteristics and behavior of polynomial functions on both real and complex planes.

Multiplicity of Zeros

Multiplicity refers to the frequency or number of times a particular zero appears as a root of the polynomial. In the case of \(P(x) = x^2 + 25\), each zero has a multiplicity of 1, because both zeros \(x = -5i\) and \(x = 5i\) come from separate linear factors \((x + 5i)\) and \((x - 5i)\).
Understanding multiplicity can provide insight into the nature of the polynomial's graph and its touches or intersections with the x-axis.

• A multiplicity of 1 indicates a simple intersection or crossing at that root.
• If the multiplicity were higher, it might imply a touch or repeated intersection at the point.

Recognizing and determining multiplicity is crucial for higher-level polynomial analysis, providing deeper insight into the structure and symmetry of polynomial graphs.