Problem 42 - Precalculus with Calculus Previews

Question

Find the zeros of the given polynomial function \(f .\) State the multiplicity of each zero. $$ f(x)=\left(x^{2}+25\right)\left(x^{2}-5 x+4\right)^{2} $$

Step-by-Step Solution

Answer

Real zero: 1, 4 (multiplicity 2); Complex zeros: 5i, -5i (multiplicity 1).

Step 1: Understanding the Polynomial

The given polynomial function is \( f(x) = (x^2 + 25)(x^2 - 5x + 4)^2 \). To find the zeros, we need to solve both \( x^2 + 25 = 0 \) and \( x^2 - 5x + 4 = 0 \).

Step 2: Finding Zeros from \(x^2 + 25\)

The equation \( x^2 + 25 = 0 \) can be rewritten as \( x^2 = -25 \). This has no real solutions as the square of a real number cannot be negative. However, we can find complex solutions: \( x = \pm 5i \). These are imaginary zeros with multiplicity 1 each.

Step 3: Finding Zeros from \(x^2 - 5x + 4\)

Factor the quadratic \( x^2 - 5x + 4 \). This can be factored into \( (x - 1)(x - 4) = 0 \). Thus, \( x = 1 \) and \( x = 4 \) are the zeros, each with a multiplicity of 2, due to the square in \( (x^2 - 5x + 4)^2 \).

Step 4: Statement of the Zeros and Their Multiplicities

The zeros of the polynomial are \( x = 1 \) and \( x = 4 \) with multiplicity 2 each. The polynomial also has complex zeros \( x = 5i \) and \( x = -5i \) with multiplicity 1 each.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are essential when solving polynomial equations that have no real solutions. The imaginary unit is represented by the symbol \( i \), where \( i^2 = -1 \). This means any number involving the square root of a negative number can be expressed using complex numbers.
For example, in the polynomial equation \( x^2 + 25 = 0 \), the solution involves imaginary numbers since \( x^2 = -25 \) has no real square roots. Instead, we find the complex solutions \( x = 5i \) and \( x = -5i \), using the fact that \( \sqrt{-25} = \pm 5i \).
Complex numbers are crucial for understanding polynomial equations in the realm of imaginary solutions, allowing students to find zeros that cannot be located on a traditional real number line.

Zero Multiplicity

Zero multiplicity refers to the number of times a particular zero appears in a polynomial function. Understanding multiplicity is essential for determining the behavior of a polynomial graph at its zeros.
When a zero has a higher multiplicity, it indicates that the function's graph touches or intersects the x-axis at that zero point more prominently. For instance, a zero with multiplicity 1 means the graph crosses the x-axis, while a zero with multiplicity 2 suggests the graph merely touches the x-axis without crossing it.
In our example, the equation \( x^2 - 5x + 4 = 0 \) generates zeros \( x = 1 \) and \( x = 4 \), each with multiplicity 2. This is due to the polynomial being squared, \( (x^2 - 5x + 4)^2 \). Thus, these zeros are repeated, indicating that the polynomial graph touches the x-axis but doesn't cross it at these points.

Factoring Polynomials

Factoring polynomials is a technique used to express a polynomial as a product of its simpler terms or factors. It is an essential tool for solving polynomial equations, as it enables easy identification of a polynomial's zeros.
To factor a polynomial like \( x^2 - 5x + 4 \), one can use various methods such as trial and error, grouping, or the quadratic formula. In our given exercise, the expression is successfully factored as \( (x - 1)(x - 4) \), identifying the zeros \( x = 1 \) and \( x = 4 \).
Factoring is not only useful for solving equations but also for simplifying expressions to better understand the function's graph and the behavior of polynomial equations. Overall, it facilitates the exploration of more complex polynomial expressions by breaking them down into manageable parts.