Problem 11 - Algebra 2

Question

Multiply. State any restrictions on the variables. $$ \frac{x^{2}-4}{x^{2}-1} \cdot \frac{x+1}{x^{2}+2 x} $$

Step-by-Step Solution

Answer

The multiplication of the given expressions simplifies to \((x-2)/(x(x-1))\) with restrictions \(x=0\) and \(x=1\).

Step 1: Factor the expressions

First, factorise the expressions. You can factorise \(x^{2}-4\) as \((x+2)(x-2)\) and \(x^{2}-1\) as \((x+1)(x-1)\). The expression \(x^{2}+2x\) can be factorised as \(x(x+2)\).

Step 2: Multiply the factored expressions

After the factorisation, let's multiply the factors. We write it as \((x+2)(x-2) / ((x+1)(x-1)) * (x+1) / x(x+2)\).

Step 3: Cancel out the common factors

By looking at the expression, we see that the factor \((x+2)\) is in both the numerator and the denominator, so they cancel each other out. We also have \((x+1)\) in both the numerator and the denominator, so they cancel each other out. The simplified expression is then \((x-2)/(x(x-1))\).

Step 4: Identify the Restrictions

The restrictions for the variables are the values that will make the denominator equal to zero because division by zero is undefined. We therefore set both \(x\) and \((x-1)\) equal to zero and solve for \(x\). This gives \(x=0\) and \(x=1\) as the restrictions.

Key Concepts

Factoring Quadratic Expressions

Quadratic expressions often appear in algebra, and factoring them is a crucial skill to master. When dealing with expressions such as \(x^{2} - 4\) or \(x^{2} - 1\), you should first recognize these as differences of squares. A difference of squares can be factored using the formula \(a^{2} - b^{2} = (a + b)(a - b)\).
For the expression \(x^{2} - 4\), it breaks down to \((x + 2)(x - 2)\). Similarly, \(x^{2} - 1\) becomes \((x + 1)(x - 1)\). These factorizations simplify our work in simplifying expressions.
Another form you might encounter is simple factoring of terms, such as \(x^{2} + 2x\), which factors out an \(x\) to give \(x(x + 2)\). This process reveals factors that are crucial when simplifying or solving equations.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form. A rational expression is a fraction where the numerator and the denominator are both polynomials. After factoring, the next step is to cancel common factors in both the numerator and the denominator.
In the expression we considered: \[\frac{(x+2)(x-2)}{(x+1)(x-1)} \cdot \frac{x+1}{x(x+2)}\]First, observe that \((x+2)\) and \((x+1)\) are factors that appear in both the numerator and the denominator. You can cancel these out to simplify the expression.

• Cancel \((x+2)\) from both numerator and denominator.
• Cancel \((x+1)\) from both numerator and denominator.

This leaves you with \(\frac{x-2}{x(x-1)}\), a much more manageable expression. Simplifying in this manner helps to make problems less complex and solutions easier to find.

Variable Restrictions

When working with rational expressions, it's important to determine any restrictions on the variables. A restriction is any value that would make the denominator of the expression equal to zero. Because division by zero is undefined, these values must be identified and excluded from the possible solutions or domain of the expression.
Consider the expression's denominator: originally \((x+1)(x-1)\) and \(x(x+2)\). To find the restrictions, set each factor equal to zero and solve for \(x\):

• \(x+1 = 0 \rightarrow x = -1\)
• \(x-1 = 0 \rightarrow x = 1\)
• \(x = 0\)

These solutions, \(x = -1\), \(x = 0\), and \(x = 1\), are the restrictions. They must be excluded from any solution set for the expression or equation involving this rational expression to ensure the calculation remains valid.