Problem 1 - Algebra Form and Function

Question

Find the zeros of the polynomials. $$ (x-3)(x-4)(x+2) $$

Step-by-Step Solution

Answer

Answer: The zeros of the polynomial are \(x = 3\), \(x = 4\), and \(x = -2\).

Step 1: Identify the factors

The polynomial is already factored into three factors: $$ (x-3)(x-4)(x+2) $$

Step 2: Set each factor equal to zero

We need to set each of the factors equal to zero and solve for x: $$ x-3=0 \\ x-4=0 \\ x+2=0 $$

Step 3: Solve for x in each equation

Now, we will solve each of the equations for x: 1. For the equation \(x-3=0\), add \(3\) to both sides: $$ x=3 $$ 2. For the equation \(x-4=0\), add \(4\) to both sides: $$ x=4 $$ 3. For the equation \(x+2=0\), subtract \(2\) from both sides: $$ x=-2 $$

Step 4: Write down the zeros

The zeros of the given polynomial are the values of x that we have found: $$ x=3, \ x=4, \ \text{and} \ x=-2 $$ These are the values of x for which the polynomial \((x-3)(x-4)(x+2)\) is equal to zero.

Key Concepts

Factoring Polynomials

Factoring polynomials is a key technique used in algebra that involves rewriting a polynomial as a product of its simplest parts called factors. This method simplifies the process of solving polynomial equations.
When a polynomial is factored, it is expressed as a multiplication of two or more simpler polynomials. In the original exercise, the polynomial \((x-3)(x-4)(x+2)\) is already factored into three linear factors: \((x-3)\), \((x-4)\), and \((x+2)\).

• Factored polynomials are easier to work with because they break down complex expressions into simpler parts.
• Finding factors is the first critical step in solving polynomial equations.
• Each factor represents a potential solution or zero of the polynomial.

Factoring is particularly useful because solving simpler equations is much more straightforward and enables students to analyze polynomials in terms of their roots and behavior.

Solving Equations

Solving equations involves finding values for the variables that make the equation true. In the context of polynomials, once a polynomial is fully factored, solving the equation is much more manageable.
The process includes setting each factor of the polynomial equal to zero. For instance, with the factors \((x-3)\), \((x-4)\), and \((x+2)\), we form the following equations:

• \(x-3=0\)
• \(x-4=0\)
• \(x+2=0\)

By solving these equations, we determine the values of \(x\) that satisfy each equation:

• Solving \(x-3=0\), we add 3 to both sides to get \(x=3\).
• Solving \(x-4=0\), we add 4 to both sides to get \(x=4\).
• Solving \(x+2=0\), we subtract 2 from both sides to get \(x=-2\).

These solutions are the values where the original polynomial equals zero.

Polynomial Roots

Polynomial roots are the values of the variable that make the polynomial equal to zero. They are also called zeros of the polynomial.
Understanding roots helps in analyzing and graphing polynomial functions. In our example, the polynomial \((x-3)(x-4)(x+2)\) has roots at \(x=3\), \(x=4\), and \(x=-2\).

• Roots are critical because they indicate where the graph of the polynomial will intersect the x-axis.
• Each root corresponds to one solution of the polynomial equation when set to zero.
• For a factored polynomial, each factor yields one root when set to zero.

Finding these roots is essential in many areas of algebra and calculus, as they provide insights into the behavior of polynomial functions, such as their growth and decline on the graph.