Problem 111 - Elementary Algebra

Question

Explain why \(2(x+5)(x-5)=0\) has two solutions and \(2 x(x+5)(x-5)=0\) has three solutions.

Step-by-Step Solution

Answer

The first equation has two solutions due to two factors, while the second has three solutions due to three factors.

Step 1: Analyze the equation

For the equation \(2(x+5)(x-5)=0\), identify that the product of factors \((x+5)\) and \((x-5)\) is zero, meaning each factor independently can be zero.

Step 2: Solving the first equation

To solve \(2(x+5)(x-5)=0\), set each factor equal to zero: \(x+5=0\) and \(x-5=0\). Solving these gives \(x=-5\) and \(x=5\). These are the two solutions for this equation.

Step 3: Analyze the second equation

For the equation \(2x(x+5)(x-5)=0\), identify three factors: \(x\), \((x+5)\), and \((x-5)\). Each factor being zero would resolve the equation to zero.

Step 4: Solving the second equation

Solve each factor independently: \(x=0\), \(x+5=0\), and \(x-5=0\). This results in three solutions: \(x=0\), \(x=-5\), and \(x=5\).

Step 5: Concluding the number of solutions

The first equation \(2(x+5)(x-5)=0\) only uses the factors \((x+5)\) and \((x-5)\), yielding two solutions. The second equation \(2x(x+5)(x-5)=0\) has an additional factor \(x\), leading to an extra solution.

Key Concepts

Factors of Polynomials

In algebra, a polynomial can have multiple factors, which are expressions that multiply together to form the polynomial. For example, in the polynomial equation \(2(x+5)(x-5)=0\), you can identify the factors as \((x+5)\) and \((x-5)\). These factors represent the building blocks of the polynomial.

• A factor is a component of a polynomial that can be separated without any remainder.
• If each factor is set to zero, it can provide potential solutions to the polynomial equation.
• The number of factors directly influences how many solutions you could potentially find.

By understanding the factors of a polynomial, it becomes easier to solve the equation by focusing on finding the values that make each factor zero.

Zero Product Property

The zero product property is a vital concept when solving polynomial equations. It states that if the product of two or more numbers is zero, then at least one of the numbers must be zero. In the equation \(2(x+5)(x-5)=0\), this property tells us that for the equation to hold true, either \((x+5)=0\) or \((x-5)=0\) (or both).

• The idea is straightforward: if you multiply several terms and the result is zero, one or more of the terms must be zero.
• Once you isolate a factor and set it to zero, you can solve for the variable to find a solution.
• This makes it an effective way to solve polynomial equations by breaking them down into simpler parts.

By applying this property, we can simply solve the components like an arithmetic puzzle: find the values that zero out each part of the product.

Number of Solutions in Equations

The number of solutions in an equation depends on the number of unique factors that are set to zero, thanks to the zero product property. For example, in the equation \(2(x+5)(x-5)=0\), there are two solutions derived from setting each factor to zero: \(x=-5\) and \(x=5\). Consider the equation \(2x(x+5)(x-5)=0\). Here, there's an extra factor \(x\), so:

• Setting \(x=0\) gives an additional solution. Thus, this equation has three solutions: \(x=0\), \(x=-5\), and \(x=5\).
• An increase in the number of factors generally increases the potential number of solutions.
• However, factors might repeat or lead to extraneous (non-unique) solutions, which need careful analysis.

Ultimately, the presence or absence of additional factors in an equation can change the total number of solutions analyzed through zeroing each factor independently.