Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the rational expressions. \(\frac{2 x^{2}+7 x-4}{4 x^{2}+2 x-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+4}{2(x+1)}\), with domain restrictions \(x \neq \frac{1}{2}\) and \(x \neq -1\).

Step 1: Factor the Numerator

Start by factoring the numerator of the rational expression \(2x^2 + 7x - 4\). We look for two numbers that multiply to \(2\times (-4) = -8\) and add to \(7\). These numbers are \(8\) and \(-1\). Using these numbers, split the middle term: \(2x^2 + 8x - x - 4\). Then, factor by grouping: \((2x^2 + 8x) + (-x - 4) = 2x(x + 4) - 1(x + 4)\). Therefore, the factored form is \((2x - 1)(x + 4)\).

Step 2: Factor the Denominator

Next, factor the denominator \(4x^2 + 2x - 2\). Again, look for two numbers that multiply to \(4\times (-2) = -8\) and add to \(2\). These numbers are \(4\) and \(-2\). Use these to split the middle term: \(4x^2 + 4x - 2x - 2\). Factor by grouping: \((4x^2 + 4x) + (-2x - 2) = 4x(x + 1) - 2(x + 1)\). The factored form is \((4x - 2)(x + 1)\).

Step 3: Simplify the Expression

The original expression \(\frac{2x^2 + 7x - 4}{4x^2 + 2x - 2}\) can now be written using the factored forms from the previous steps: \(\frac{(2x - 1)(x + 4)}{(4x - 2)(x + 1)}\). Notice that \(4x - 2\) can be further simplified by factoring out a 2 to get \(2(2x - 1)\). This gives us: \(\frac{(2x - 1)(x + 4)}{2(2x - 1)(x + 1)}\). Cancel \(2x - 1\) from the numerator and denominator. The simplified expression is \(\frac{x + 4}{2(x + 1)}\).

Step 4: State the Domain Restrictions

Since we canceled \(2x - 1\) and there was an \(x + 1\) in the denominator, ensure that these don’t equate to zero to avoid undefined expressions. Thus, set \(2x - 1 eq 0\) (i.e., \(x eq \frac{1}{2}\)) and \(x + 1 eq 0\) (i.e., \(x eq -1\)). The domain of the original expression is all real numbers except \(x = \frac{1}{2}\) and \(x = -1\).

Key Concepts

Factoring in Rational Expressions

Factoring is the process of rewriting an expression as a product of its factors. It is a crucial skill when dealing with rational expressions because it allows us to simplify them effectively. For our exercise, we start by factoring both the numerator and the denominator.

**Numerator**: We have the expression \(2x^2 + 7x - 4\). To factor this expression, we look for two numbers whose product equals \(-8\) (which is \(2 \times -4\)) and that sum up to \(7\). These numbers are \(8\) and \(-1\). Using these numbers, we can rewrite the middle term, which results in:

• \(2x^2 + 8x - x - 4\)
• Group terms to factor: \(2x(x + 4) - 1(x + 4)\)

This results in the factored form: \((2x - 1)(x + 4)\).

**Denominator**: For \(4x^2 + 2x - 2\), we follow a similar process. The numbers needed multiply to \(-8\) and add to \(2\), these are \(4\) and \(-2\). Split the middle term:

• \(4x^2 + 4x - 2x - 2\)
• Group to factor: \(4x(x + 1) - 2(x + 1)\)

This gives the factored form: \((4x - 2)(x + 1)\). Factoring correctly is essential to simplifying rational expressions accurately.

Simplification of Rational Expressions

Once the expressions have been factored, simplifying them becomes much more straightforward. Simplification primarily involves canceling out common factors from the numerator and the denominator.

With our exercise, the expression \(\frac{2x^2 + 7x - 4}{4x^2 + 2x - 2}\) can be rewritten using the factors found:

• Numerator: \((2x - 1)(x + 4)\)
• Denominator: \((4x - 2)(x + 1)\)

Notice that \(4x - 2\) can further be simplified by factoring out a 2, yielding \(2(2x - 1)\). This step is key:

• \(\frac{(2x - 1)(x + 4)}{2(2x - 1)(x + 1)}\)

In this form, you can see that \((2x - 1)\) is a common factor in both the numerator and the denominator. By canceling \((2x - 1)\), we simplify the expression to \(\frac{x + 4}{2(x + 1)}\).

Remember, always look for common factors to cancel in simplifying, and ensure they are factored out correctly before cancelation.

Understanding Domain Restrictions

Domain restrictions in rational expressions determine the values that \(x\) cannot take, to ensure that the expression remains defined. Remember, a rational expression is undefined wherever the denominator equals zero.

In our problem, the original denominator \(4x^2 + 2x - 2\) was factored to \((4x - 2)(x + 1)\). Before simplification, we set each factor in the denominator equal to zero to find the restrictions:

• \(4x - 2 = 0 \Rightarrow x = \frac{1}{2}\)
• \(x + 1 = 0 \Rightarrow x = -1\)

These calculations mean that if \(x = \frac{1}{2}\) or \(x = -1\), the denominator becomes zero, and the expression is undefined.

Even after simplifying, we must retain these restrictions because they reflect the constraints of the original expression. Therefore, the domain of the rational expression experience includes all real numbers except \(x = \frac{1}{2}\) and \(x = -1\). Recognizing these restrictions is crucial for fully understanding the function of rational expressions in various mathematical contexts.