Problem 10 - Intermediate Algebra

Question

A student checks her answers with those in the back of her textbook. Determine whether they are equivalent. $$ \begin{array}{|c|c|c|} \hline \text { Student's answer } & \text { Book's answer } & \text { Equivalent? } \\ \hline \frac{3+2 t}{t^{2}+2 t} & \frac{2 t+3}{t(t+2)} & \\ \hline \frac{5-3 x^{2}}{x+x^{2}} & -\frac{3 x^{2}-5}{x^{2}+x} & \\ \hline \frac{3 x y(y+x)}{(2 y-x)(2 y+3 x)} & \frac{3 x y^{2}+3 x^{2} y}{(2 y+x)(2 y-3 x)} & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

First two answers are equivalent; the third is not.

Step 1: Compare the First Set of Answers

We first consider the expression \( \frac{3+2t}{t^{2}+2t} \) and attempt to simplify it. Factor the denominator: \( t^2 + 2t = t(t+2) \). So, the expression becomes \( \frac{3+2t}{t(t+2)} \). Comparing this to the book's answer \( \frac{2t+3}{t(t+2)} \), the forms are identical. **Yes, they are equivalent.**

Step 2: Compare the Second Set of Answers

The student's answer is \( \frac{5-3x^2}{x+x^2} \), and we rewrite it as \( \frac{5 - 3x^2}{x(x+1)} \). The book's answer is \( -\frac{3x^2-5}{x^2+x} = \frac{5-3x^2}{x(x+1)} \) after factoring a negative out of the numerator. **Yes, they are equivalent.**

Step 3: Compare the Third Set of Answers

For the expression \( \frac{3xy(y+x)}{(2y-x)(2y+3x)} \), expand the numerator: \( 3x(y^2 + xy) \). This matches the book's expansion \( \frac{3xy^2 + 3x^2y}{(2y+x)(2y-3x)} \). The denominators \( (2y-x)(2y+3x) \) and \( (2y+x)(2y-3x) \) are different, as these are different factors. **No, they are not equivalent.**

Key Concepts

Simplification of Rational Expressions

Simplifying rational expressions is akin to simplifying fractions in basic arithmetic. Just like you simplify \( \frac{6}{8} \) to \( \frac{3}{4} \), rational expressions can be simplified by canceling out common factors in the numerator and the denominator. To simplify a rational expression:

• Factor both the numerator and the denominator completely.
• Identify and cancel out the common factors.

Let's take an example from our exercise: The expression \( \frac{3+2t}{t^2 + 2t} \) becomes \( \frac{3+2t}{t(t+2)} \) once you factor the denominator. You can now see that it's equivalent to \( \frac{2t+3}{t(t+2)} \), showing that the two versions are indeed equivalent.
The key is understanding how factoring works here and applying it correctly to cancel terms effectively.

Factoring Polynomials

Factoring is breaking down a polynomial into a product of simpler terms or polynomials. This is crucial in simplifying expressions and solving equations. The fundamental goal is to express a polynomial as a product of other polynomials that cannot be factored further. Here's how to factor basic polynomials:

• Identify the greatest common factor (GCF) of the polynomial terms.
• Rewrite the polynomial as a product of the GCF and another polynomial.

For example, consider the polynomial \( x^2 + x \). The GCF is \( x \), thus you can factor this polynomial as \( x(x + 1) \).
For more complex terms like \( 3x^2 - 5 \), if you factor out a negative, you get: \( -1(3x^2 - 5) \). This step was crucial in the second set of expressions in our exercise.Use the right techniques, like splitting the middle term or using synthetic division, when dealing with higher-degree polynomials.

Comparison of Expressions

To determine if two rational expressions are equivalent, we primarily focus on their form after simplification. Here's a simple approach to comparing expressions:

• Ensure both expressions are fully simplified and factored.
• Check if both final forms match exactly.

In our exercise, observe how the expression \( \frac{5-3x^2}{x(x+1)} \) matched the book’s expression when simplified and compared carefully, demonstrating equivalency.
However, if during simplification, the resulting expressions have different factors or do not match, like the third example, the expressions are not equivalent.The process demands attention to detail in factoring and simplification steps, ensuring that equivalent expressions remain consistent across different transformations.