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Chapter 6: Problem 31

Write each rational expression in lowest terms. $$ \frac{(x+4)(x-3)}{(x+5)(x+4)} $$

Short Answer

Expert verified
\(\frac{x-3}{x+5}\)

Step by step solution

01

Factor the numerator and denominator

Identify any factors in the numerator and the denominator. In this case, the numerator is \( (x+4)(x-3) \) and the denominator is \( (x+5)(x+4) \). Confirm if there are any common factors.
02

Cancel the common factors

Both the numerator and the denominator have a common factor of \( x+4 \). Divide the numerator and the denominator by this common factor: \[ \frac{(x+4)(x-3)}{(x+5)(x+4)} = \frac{(x-3)}{(x+5)} \].
03

Simplify the expression

After canceling the common factor, you're left with the simplified expression \( \frac{x-3}{x+5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials means breaking them down into simpler expressions that can be multiplied together to get the original polynomial. It is much like breaking down numbers. For example, the number 12 can be factored into 3 and 4 because 3 multiplied by 4 equals 12. In our exercise, the polynomial in the numerator is \( (x+4)(x-3) \) and in the denominator is \( (x+5)(x+4) \). Both are already factored for us.

If they weren't factored, we would need to find a way to write them in this form. For instance, if we had \( x^2 + x - 12 \), we could factor it into \( (x+4)(x-3) \). Understanding how to factor polynomials helps you see the hidden elements that are multiplied together.
Common Factors
A common factor is a number or expression that is a factor of two or more numbers or expressions. In the fraction \( \frac{(x+4)(x-3)}{(x+5)(x+4)} \), \( (x+4) \) appears in both the numerator and the denominator. This means \( (x+4) \) is a common factor and can be canceled out.

To cancel a common factor, you divide both the numerator and the denominator by that factor. This process uses the principle of simplification, where you divide out anything that is the same in both the top and bottom of a fraction. Just as you would cancel the 3s in \( \frac{3 \times 8}{3 \times 2} \), you're canceling \( (x+4) \) in this problem.
Simplified Expressions
A simplified expression is one that has been reduced to its lowest terms. It鈥檚 easier to understand and usually reveals more about the nature of the function. In our exercise, we start with the fraction \( \frac{(x+4)(x-3)}{(x+5)(x+4)} \). After canceling the common factor of \( (x+4) \), we are left with \( \frac{x-3}{x+5} \).

This expression can鈥檛 be simplified any further because there are no more common factors to cancel. Simplified expressions are valuable in math because they provide a clean and easy-to-use form of an equation or function. It helps in solving algebraic problems more efficiently by reducing complexity.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{t^{2}-9}{3 t+9} $$

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