Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation. \((x+1)(x+2)(x-4)=0\)

Step-by-Step Solution

Answer

The solutions for the equation are \(x=-1\), \(x=-2\), \(x=4\)

Step 1: Set first factor equal to zero

The first factor is \(x+1\). Set this factor equal to zero and solve for \(x\): \[x+1=0\] Subtract 1 from both sides to obtain: \[x=-1\]

Step 2: Set second factor equal to zero

The second factor is \(x+2\). Set this factor equal to zero and solve for \(x\): \[x+2=0\] Subtract 2 from both sides to obtain: \[x=-2\]

Step 3: Set third factor equal to zero

The third factor is \(x-4\). Set this factor equal to zero and solve for \(x\): \[x-4=0\] Add 4 to both sides to obtain: \[x=4\]

Key Concepts

Factorization

Factorization is a crucial method in solving quadratic equations. In algebra, it involves expressing a polynomial as a product of its factors, or simpler expressions. By breaking down the equation into these variables, you can solve it piecemeal.
For example, with the equation \((x+1)(x+2)(x-4)=0\), each part within the parentheses is a factor of the equation. A factor is essentially a component that, when multiplied by the other factors, results in the original expression.
The factored form is more useful than expanding the entire equation, especially when applying the zero-product property that relies on these individual terms. To identify the factors, one often looks for patterns or uses techniques such as the Greatest Common Factor (GCF) or trial and error when the terms are more complex. Preparation and identification of the factors can simplify the solving process considerably.

Zero-Product Property

The zero-product property is a principle that greatly simplifies solving polynomial equations, such as our given equation \((x+1)(x+2)(x-4)=0\). This mathematical rule states that if the product of two or more factors equals zero, then at least one of the factors must be zero.
In simple terms, if \(ab = 0\), then \(a = 0\), \(b = 0\), or both. This dynamic allows us to set each factor in a factored polynomial equal to zero and solve for the variable.

• For \(x+1=0\), solving gives \(x = -1\).
• For \(x+2=0\), solving gives \(x = -2\).
• For \(x-4=0\), solving gives \(x = 4\).

This approach efficiently narrows down potential solutions without needing to expand or make complex calculations of the entire polynomial equation.

Roots of Equations

Finding the roots of an equation means determining the values of the variable that satisfy the equation, making the overall expression equal to zero. These roots can also be called solutions or zeros. In the context of polynomial functions, a root tells you where the graph intersects the horizontal axis when graphed on a coordinate plane.
From the exercise \((x+1)(x+2)(x-4)=0\), our calculations showed that the roots are \(x = -1\), \(x = -2\), and \(x = 4\). Each of these represents a different point where the equation equals zero, and therefore, where the graph of the equation would touch the x-axis.

• Roots are critical in numerous real-world applications, such as physics for determining motion or economics for understanding equilibrium points.
• They provide key insights into the behavior of functions and are fundamental in calculus and other areas of advanced mathematics.

Understanding the roots of equations allows you to solve practical problems and provides a deeper insight into the structure of mathematical expressions.