Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 7 - Rational Functions - 7.4 Adding and Subtracting Rational Expressions - 7.4 Exercises - Page 589: 19

Answer

LCD = $(t - 7)(t + 5)(t - 4)$ $\frac{(t + 3)(t - 4)}{(t - 7)(t + 5)(t - 4)}$ $\frac{(t - 7)(t - 7)}{(t - 7)(t + 5)(t - 4)}$

Work Step by Step

The first thing we want to do is to make sure that each denominator is factored completely. We see that the denominators are quadratic expressions, which are given by the formula: $ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers. To factor the expression in the denominator of the first fraction, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $-35$, but when added together will give us the $b$ term, which is $-2$. This means that one factor is positive and the other is negative, but the negative factor has the greater absolute value. Let's look at possible factors: $-7$ and $5$ $-35$ and $1$ It looks like the first combination will work. Let's rewrite the expression in factor form: $(t - 7)(t + 5)$ To factor the expression in the denominator of the second fraction, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $-20$, but when added together will give us the $b$ term, which is $1$. This means that one factor is positive and the other is negative, but the positive factor has the greater absolute value. Let's look at possible factors: $5$ and $-4$ $10$ and $-2$ $20$ and $-1$ It looks like the first combination will work. Let's rewrite the expression in factor form: $(t + 5)(t - 4)$ The two fractions can now be rewritten as: $\frac{t + 3}{(t - 7)(t + 5)}$ $\frac{t - 7}{(t + 5)(t - 4)}$ Now, we want to find the least common denominator (LCD). We do this by taking the highest power of each factor in the denominators of the fractions: LCD = $(t - 7)(t + 5)(t - 4)$ Now that we have the least common denominator, we multiply the numerator of each fraction with the factor or factors it is missing in its denominator: $\frac{(t + 3)(t - 4)}{(t - 7)(t + 5)(t - 4)}$ $\frac{(t - 7)(t - 7)}{(t - 7)(t + 5)(t - 4)}$
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