Problem 6 - Precalculus

Question

Determine whether the given value for the variable is a root of the equation. $$(x-1)(x-2)(x-3)=0 ; x=4$$

Step-by-Step Solution

Answer

No, \(x = 4\) is not a root of the equation.

Step 1: Understanding the Equation

The equation given is \((x-1)(x-2)(x-3)=0\). For \(x\) to be a root of this equation, it must satisfy the equation, meaning at least one of the factors \((x-1)\), \((x-2)\), or \((x-3)\) should equal zero.

Step 2: Substituting the Given Value

We need to check if \(x = 4\) is a root. Substitute 4 into each factor of the equation: \((4-1)\), \((4-2)\), and \((4-3)\).

Step 3: Evaluating Each Factor

Calculate each factor: \(4-1 = 3\), \(4-2 = 2\), and \(4-3 = 1\). None of these results equal zero.

Step 4: Determining the Result

Since none of the factors equaled zero, \(x=4\) does not satisfy the equation \((x-1)(x-2)(x-3)=0\). Therefore, \(x=4\) is not a root of the equation.

Key Concepts

Factoring Polynomials

Factoring polynomials is an important concept in algebra that allows us to break down complex polynomial expressions into simpler ones. This is useful when solving equations, like in the given problem. When you have an equation such as \[(x-1)(x-2)(x-3)=0,\]factoring helps identify the values of \(x\) that can make the product zero. Polynomials can often be factored into monomials (single term algebraic expressions) or binomials. In the context of the given exercise, the polynomial \[(x-1)(x-2)(x-3)\]is already presented in factored form. This indicates that each factor could potentially be zero, thus determining the roots of the polynomial.

• The aim is to isolate values of \(x\) that cause any one of these factors to equal zero.
• When a factor equals zero, the entire equation results in zero.

This concept is crucial because it provides a straightforward mechanism for identifying roots, which are essentially the solutions to the equation. Being able to factor polynomials efficiently can significantly simplify the process of solving polynomial equations.

Evaluating Expressions

Evaluating expressions involves substituting specific values into a polynomial or algebraic equation to determine the outcome. This step is essential when checking if a particular number is a root of the equation.In the problem \[(x-1)(x-2)(x-3)=0,\]we need to evaluate each expression \[(x-1), (x-2),\]and \[(x-3)\]based on the given value \(x=4\). By substituting \(x=4\) into each factor, we calculate

• \(4-1 = 3\)
• \(4-2 = 2\)
• \(4-3 = 1\)

Evaluating these expressions shows that none of the factors equal zero. This means that substituting \(x=4\) into the polynomial does not yield a result of zero for any of the factors, confirming that \(x=4\) is not a root of the equation. Understanding how to evaluate expressions is vital as it helps verify potential solutions by systematically testing variable values within the equation.

Equation Solving

Equation solving is a core skill that combines many mathematical techniques to find the values, or roots, that make an equation true. The provided problem \[(x-1)(x-2)(x-3)=0,\]requires finding values of \(x\) that can make the entire expression equal to zero. This technique is part of the zero-product property, which states that if a product of multiple factors equals zero, then at least one factor must be zero.

• In this scenario: \(x-1=0\), \(x-2=0\), or \(x-3=0\).

Solving these equations individually gives potential roots for \(x\):

• \(x = 1\)
• \(x = 2\)
• \(x = 3\)

By applying equation solving, it becomes clear that when \(x=4\), none of the factors results in zero, thus confirming \(x=4\) is not a root.Mastering equation solving is important because it is foundational not just for algebra, but for various applications in mathematics and sciences. It enables tackling complex equations methodically to arrive at correct solutions.