Problem 42 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{3+x-2 x^{2}}{2+x-x^{2}}\)

Step-by-Step Solution

Answer

Expression is already simplified.

Step 1: Rearrange terms in descending order

The numerator of the expression is \(3 + x - 2x^2\). Rearrange it to \(-2x^2 + x + 3\). Similarly, rearrange the denominator \(2 + x - x^2\) to \(-x^2 + x + 2\).

Step 2: Factor the numerator

Consider the expression \(-2x^2 + x + 3\). To factor, we seek two numbers whose product is \(-6\) (the product of \(a = -2\) and \(c = 3\)) and whose sum is \(1\) (the coefficient of \(x\)). These numbers are \(3\) and \(-2\). Therefore, the numerator can be factored as \((-2x^2 + 3x - 2x + 3)\).

Step 3: Factor numerator completely

Group the terms in the numerator: \([(-2x^2 + 3x) + (-2x + 3)]\). Factor by grouping: \(-x(2x - 3) - 1(2x - 3)\). This results in \((-x - 1)(2x - 3)\).

Step 4: Factor the denominator

Consider the expression \(-x^2 + x + 2\). To factor, find two numbers whose product is \(-2\) (the product of \(a = -1\) and \(c = 2\)) and whose sum is \(1\) (the coefficient of \(x\)). These numbers are \(2\) and \(-1\). Thus, the denominator can be factored as \((-x^2 + 2x - x + 2)\).

Step 5: Factor denominator completely

Group the terms in the denominator: \([(-x^2 + 2x) + (-x + 2)]\). Factor by grouping: \(-x(x - 2) + 1(x - 2)\). This simplifies to \((-x + 1)(x - 2)\).

Step 6: Cancel common factors

The factored form of the expression is \(\frac{(-x - 1)(2x - 3)}{(-x + 1)(x - 2)}\). There are no common factors in both the numerator and the denominator to cancel.

Key Concepts

Factoring

Factoring is a method where we break down an algebraic expression into the product of its simpler elements or factors. This is especially useful in simplifying rational expressions. For a quadratic expression like \(-2x^2 + x + 3\), our goal is to express it as a product of two binomials.

We do this by identifying two numbers which multiply to give the product of the coefficient of \(x^2\) (denoted as \(a\)) and the constant term (denoted as \(c\). For \(-2x^2 + x + 3\), we find two numbers that multiply to \(-6\) (since \(-2 imes 3 = -6\)) and add up to \(1\). These numbers are \(3\) and \(-2\).

This lets us rewrite and factor the expression as \(-2x^2 + 3x - 2x + 3\) followed by grouping and factoring by parts: \((-x - 1)(2x - 3)\). The same process applies to the denominator. Factoring helps in simplifying expressions by revealing common factors to be canceled out.

Numerator and Denominator

In a rational expression, the numerator is the top part and the denominator is the bottom part. Understanding each part is crucial when simplifying these types of expressions.

• The numerator of our expression, \(3 + x - 2x^2\), is rewritten in standard form as \(-2x^2 + x + 3\). This involves arranging the terms starting with the highest power of \(x\).
• Similarly, the denominator \(2 + x - x^2\) is rewritten in standard form as \(-x^2 + x + 2\).

This restructuring is important for determining factors. It allows us to apply the factoring method described earlier in a clear manner. Without proper ordering, it would be difficult to apply systematic factoring, eventually complicating the simplification process. So, ensuring both the numerator and the denominator are in descending order of exponent helps in factor grouping.

Simplification Steps

After factoring both the numerator and the denominator completely, we aim to simplify the rational expression by canceling out common factors. Since both parts are now products of factors, we check for any shared components.

• From the problem, \((2x - 3)\) and \(x - 2)\) are distinct, meaning there are no shared factors between the numerator \((-x - 1)(2x - 3)\) and the denominator \((-x + 1)(x - 2)\).
• Without any common factors, this expression doesn't reduce further. It's critical to ensure the correct factoring and simplification process since sometimes the simplest expressions involve no cancelations.

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