Problem 5 - Elementary Algebra

Question

For the following problems, solve the equations. $$ (m-8)(m+1)=0 $$

Step-by-Step Solution

Answer

Question: Find the values of 'm' which satisfy the equation \((m-8)(m+1)=0\). Answer: The values of 'm' which satisfy the given equation are m=8 and m=-1.

Step 1: Identify the Zero-Product Property

The zero-product property states that if the product of two factors is zero, at least one of the factors must be equal to zero. In other words, if ab=0, then a=0 or b=0. In our equation, we have two factors (m-8) and (m+1). So, we will equate each factor individually to zero and solve for 'm'.

Step 2: Equate Each Factor to Zero

We have two factors: (m-8) and (m+1). Equate each factor to zero: $$ m-8=0 $$ and $$ m+1=0 $$

Step 3: Solve for 'm'

Now, we'll solve each equation for 'm'. For the first equation: $$ m-8=0 \Rightarrow m=8 $$ For the second equation: $$ m+1=0 \Rightarrow m=-1 $$

Step 4: Write Down the Solution

The solutions to the given equation are m=8 and m=-1. The equation \((m-8)(m+1)=0\) is satisfied for these values of 'm'.

Key Concepts

Zero-Product Property

The Zero-Product Property is a fundamental principle in algebra. It states that if the product of two or more factors is zero, at least one of the factors must be zero. This is an incredibly useful property for solving quadratic equations because it allows us to break down complicated expressions into simpler parts.
For example, if we have an equation like \(a \cdot b = 0\), either \(a=0\) or \(b=0\) must be true. This property gives us a direct route to finding the roots of polynomial equations.
This property is especially helpful because it reduces the complexity of solving equations. Instead of handling a product as a whole, you can focus on solving smaller, simpler equations. In our example, \( (m-8)(m+1)=0 \), we have two factors, \( m-8 \) and \( m+1 \). Using the zero-product property, we know that solving for \( m \) starts with equating each factor to zero.

Factoring Polynomials

Factoring Polynomials is a critical skill when dealing with algebraic expressions, especially quadratic equations. To factor a polynomial is to express it as a product of its factors, and when it's in a form that can be analyzed using the zero-product property. Polynomials often can be rewritten as simpler binomials or monomials.Let's see how this works with our specific example, \( (m-8)(m+1)=0 \). In this instance, the expression has already been nicely factored for us; it is given as the product of two factors: (m-8) and (m+1). Each of these is a binomial.

• The first binomial here is (m-8). It's a factor because there is no multiplication or addition inside, it is just the expression itself.
• The second binomial is (m+1). This is also a simple factor for the same reason.

So whenever you encounter a polynomial, always check if you can factor it. It might already be expressed by its factors like in our problem, or you might need to do some work to factor it.

Solving Equations

Solving Equations involves finding the value or values of variables that satisfy the equation. Simple linear equations involve one operation, while quadratic equations require methods like factoring, completing the square, or using the quadratic formula.
When solving equations through factoring, we rely heavily on the zero-product property. Let's solve the example given: \( (m-8)(m+1)=0 \).

• Start by using the zero-product property. Each individual factor must equal zero for the equation to hold true.
• We equate both factors to zero: \( m-8=0 \) and \( m+1=0 \).
• Solving each gives us the roots of the equation: \( m= 8 \) and \( m=-1 \).

Thus, the solutions to the equation are both possible values for \( m \) that satisfy \( (m-8)(m+1)=0 \). This process shows how breaking down the equation into simpler parts can guide us to the right solutions.