Problem 51 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+1)(x-1) $$

Step-by-Step Solution

Answer

Answer: x^2 - 1

Step 1: Use the distributive property

To begin, we can distribute the first term in the first binomial to both terms in the second binomial: \(x(x-1) + 1(x-1)\).

Step 2: Distribute the terms further

Now, we can distribute the terms inside the parentheses: \((x^2 - x) + (x - 1)\).

Step 3: Combine like terms

Finally, we can add the like terms in our expression: \((x^2 - x) + (x - 1) = x^2 - x + x - 1 = x^2 - 1\). Thus the product of the given expression \((x+1)(x-1)\) is \(x^2 - 1\).

Key Concepts

Distributive Property

Understanding the distributive property is fundamental when dealing with algebraic expressions, including the factoring of binomials. This property allows us to multiply a single term by each term within a parenthesis, simplifying the expression.

For instance, if we have \(a(b + c)\), we apply the distributive property by multiplying \(a\) with \(b\) and then \(a\) with \(c\), resulting in \(ab + ac\). In the exercise, the distributive property is used when the term \(x\) is multiplied by both \(x\) and \( -1\) from the binomial \(x-1\), and similarly when \(1\) is distributed across \(x\) and \( -1\).

Let's break it down:

• The term \(x\) is multiplied by \(x\), giving us \(x^2\).
• Then \(x\) is multiplied by \( -1\), yielding \( -x\).
• Following that, \(1\) is multiplied by \(x\), resulting in \(+x\).
• Finally, \(1\) is multiplied by \( -1\), giving us \( -1\).

By applying the distributive property, we transformed the product of two binomials into a simple algebraic expression \(x^2 - x + x - 1\).

Like Terms

In algebra, like terms are terms that have the exact same variables raised to the same powers, even though their coefficients may differ. The concept of like terms is crucial when simplifying algebraic expressions because we can combine them to make the expression neater and more concise.

Let's consider the expression \(x^2 - x + x - 1\) from our exercise. Notice that \( -x\) and \(+x\) are like terms because they both contain the variable \(x\) raised to the first power. When they are combined, they effectively cancel each other out, since \( -x + x = 0\). The resulting expression is then \(x^2 - 1\), which is much simpler and is the final product of the given exercise.

To identify like terms quickly, look for terms that share common variables and exponents and remember that only the coefficients can differ. When you find like terms, add or subtract them based upon their coefficients to combine them into a single term.

Polynomial Multiplication

When working with algebra, polynomial multiplication is a frequent task. Polynomials are expressions with multiple terms, which we often need to multiply together. In the context of our exercise, we have two binomials, which are essentially the simplest forms of polynomials, having two terms each.

The method we use to multiply the binomials \(x+1\) and \(x-1\) is an application of polynomial multiplication. The strategy is systematic:

• Each term in the first polynomial is multiplied by each term in the second polynomial.
• Afterward, the resulting terms are combined using the distributive property.
• Finally, any like terms are combined to simplify the expression even further.

The product of multiplying two binomial polynomials is a trinomial, which can often be simplified to a binomial or even further, as in our case, resulting in \(x^2 - 1\). Polynomial multiplication is vital for progressing in algebra, and understanding how to do it with binomials sets the foundation for dealing with more complex expressions.