91Ó°ÊÓ

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, the polynomial function given in this exercise is
\( f(x) = x^2 - 3x + 5 \)
It consists of three terms:

• The term \( x^2 \), where the coefficient is 1
• The term \( -3x \), where the coefficient is -3
• The constant term 5

The general form of a polynomial function is: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] Here, the highest power of the variable (n) is called the degree of the polynomial. The coefficients are real or complex numbers. Polynomial functions can be used to model various real-world phenomena, such as projectile motion and population growth.

Complex numbers are numbers that have both a real part and an imaginary part. They are usually expressed in the form \( a + bi \), where

• \( a \) is the real part
• \( b \) is the imaginary part, and
• \( i \) is the imaginary unit, which satisfies the equation \( i^2 = -1 \)

In the exercise, the number \( k = 1 - 2i \) is given as a complex number. Here,

• The real part is 1
• The imaginary part is -2

Complex numbers are used extensively in various fields of science and engineering. They are crucial in solving certain polynomial equations that have no real solutions.

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation \( f(x) = 0 \). Zeros are also known as roots of the polynomial. They can be real or complex numbers.

To find zeros, we can use various methods such as factoring, using the quadratic formula, or synthetic division. In this exercise, we utilize synthetic division to determine if a given number, \( k = 1 - 2i \), is a zero of the polynomial \( f(x) = x^2 - 3x + 5 \). If the remainder after synthetic division is zero, then \( k \) is a zero of the polynomial. Otherwise, the remainder represents \( f(k) \), the value of the polynomial evaluated at \( k \).

Synthetic division is a simplified form of polynomial division, especially useful for dividing by linear terms. Here are the steps:
1. **Identify the Coefficients**
For the polynomial \( f(x) = x^2 - 3x + 5 \), the coefficients are 1, -3, and 5.

2. **Set Up the Division**
Write the given \( k = 1 - 2i \) on the left and the coefficients on the top row.

3. **Carry Down the First Coefficient**
Bring down the first coefficient (1) below the line.

4. **Multiply and Add**
Multiply the carried down value by \( k \) and write the result under the next coefficient, then add. Repeat to the last coefficient. \[ \begin{array}{r|lll} & 1 & -3 & 5 \ 1-2i & & 1 & \ & (1-2i) & -3+(1-2i) & \ & & & (1-2i)[-2(1-2i)+5] \ \end{array} \]
5. **Simplify**
Simplify all terms. If the final value is zero, then \( k \) is a zero; otherwise, it is the value of \( f(k) \).

These steps make synthetic division an efficient method for verifying polynomial zeros.