Problem 7 - Algebra Form and Function

Question

Give all the solutions of the equations. $$ (x-1)(x+2)(x-3)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(x=1\), \(x=-2\), and \(x=3\).

Step 1: Write down the equation

Write the given equation: $$ (x-1)(x+2)(x-3) = 0 $$

Step 2: Identify the binomials

In the equation, we have three binomials: \((x-1)\), \((x+2)\), and \((x-3)\). For the product of these binomials to equal zero, at least one of these binomials must equal zero.

Step 3: Find the solutions

Determine the values of \(x\) that make each binomial equal to zero: For \((x-1)=0\), we have: $$ x-1=0 \Rightarrow x=1 $$ For \((x+2)=0\), we have: $$ x+2=0 \Rightarrow x=-2 $$ For \((x-3)=0\), we have: $$ x-3=0 \Rightarrow x=3 $$

Step 4: State the solutions

The solutions of the equation \((x-1)(x+2)(x-3)=0\) are \(x=1\), \(x=-2\), and \(x=3\).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental principle in algebra. It states that if the product of multiple factors equals zero, then at least one of these factors must also equal zero. This property is extremely useful when solving polynomial equations because it allows us to break down complex expressions into simpler, manageable parts.
For example, consider the equation \[(x-1)(x+2)(x-3) = 0\]Using the Zero Product Property, we can determine that at least one of these individual binomials

• \((x-1)\)
• \((x+2)\)
• \((x-3)\)

must be zero. This strategy simplifies finding the solutions for the equation by focusing on each binomial separately.

Polynomial Equations

Polynomial equations are equations consisting of variables raised to various powers, combined by addition, subtraction, and multiplication. These equations play a crucial role in mathematics, from basic algebra to advanced calculus.
The given equation\[(x-1)(x+2)(x-3) = 0\]is a polynomial equation of degree three. Every polynomial of degree three can have up to three solutions. Factoring the polynomial, as shown, helps in identifying these potential solutions.
Solving polynomial equations often involves converting the polynomial into simpler forms, such as products of binomials or factoring it into its simplest terms. This allows us to apply properties like the Zero Product Property, making it easier to identify solutions.

Binomial Solutions

Binomial solutions refer to finding the solutions for each binomial factor in a polynomial equation. A binomial is a polynomial with two terms, such as the expressions

• \((x-1)\)
• \((x+2)\)
• \((x-3)\)

from our equation.
To find the solutions for each binomial, set each equal to zero and solve for the variable \(x\). For instance, solving \[x-1=0\]results in \(x=1\). Similarly, solve the other binomials by setting

• \(x+2=0\) to get \(x=-2\)
• \(x-3=0\) to get \(x=3\)

By solving each binomial, you find the values of \(x\) that make the whole polynomial zero. Thus, the solutions for the polynomial equation are \(x=1\), \(x=-2\), and \(x=3\). This approach simplifies the process, especially for higher-degree polynomials.