Problem 96 - Elementary Algebra

Question

Solve. $$ (x-1)(x-2)(x-3)=0 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

The equation \((x-1)(x-2)(x-3)=0\) is a polynomial equation where the product of three linear factors equals zero.

Step 2: Apply the Zero Product Property

According to the Zero Product Property, if the product of multiple factors equals zero, then at least one of the factors must be zero.

Step 3: Set each factor to zero and solve

Set each linear factor \((x-1)=0\), \((x-2)=0\), and \((x-3)=0\) separately to find the values of \(x\) that make any factor (and thus the product) equal zero.

Step 4: Solve the equation \(x-1=0\)

Solve for \(x\) by adding 1 to both sides, giving \(x=1\).

Step 5: Solve the equation \(x-2=0\)

Solve for \(x\) by adding 2 to both sides, giving \(x=2\).

Step 6: Solve the equation \(x-3=0\)

Solve for \(x\) by adding 3 to both sides, giving \(x=3\).

Step 7: List the solutions

The solutions for 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 foundational concept in algebra that helps solve equations involving the product of factors equal to zero. It states that if the product of two or more numbers (or expressions) is zero, then at least one of the numbers must be zero. This property is especially useful when dealing with polynomial equations, such as

• \[ (x-1)(x-2)(x-3)=0 \]

This means we can take each factor in the equation separately and set them equal to zero. Solving each smaller equation allows us to find potential solutions to the entire polynomial equation. This process simplifies finding solutions because determining when a product of numbers is zero is straightforward.

Linear Factors

Linear factors are expressions of the first degree, such as \(x - a\), where \(a\) is a constant. These factors are straightforward because they only involve one variable raised to the first power. Each linear factor represents a straight line when graphed and simplifies solving equations by turning polynomials into manageable parts.In the equation

• \[ (x-1)(x-2)(x-3)=0 \]

we have three linear factors: \(x-1\), \(x-2\), and \(x-3\). Each of these can be solved independently. This allows for a step-by-step approach where each factor simplifies to finding a single solution for \(x\). By breaking down a polynomial equation into its linear factors, we enable an easier path to identifying all the roots.

Roots of Equations

The roots of an equation are the values that make the equation true, meaning the polynomial equals zero. Finding these roots is synonymous with solving the equation because it involves determining the values of \(x\) that satisfy the condition.For the polynomial

• \[ (x-1)(x-2)(x-3)=0 \]

we solved each linear factor by setting them equal to zero and calculating:

• \(x-1=0\) results in \(x=1\)
• \(x-2=0\) results in \(x=2\)
• \(x-3=0\) results in \(x=3\)

These solutions, \(x=1\), \(x=2\), and \(x=3\), are the roots of the polynomial equation. By identifying the roots, we pinpoint where the graph of the polynomial touches the x-axis, providing a visual understanding of the solution.