Problem 93
Question
For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Simplify: \(\frac{x^{2}}{x^{2}-5 x-24}-\frac{5 x+40}{x^{2}-5 x-24}\) Incorrect Answer: \(\frac{x^{2}}{x^{2}-5 x-24}-\frac{3 x+40}{x^{2}-5 x-24}\) $$ =\frac{x^{2}-3 x+40}{x^{2}-5 x-24} $$
Step-by-Step Solution
Verified Answer
The mistake was in the numerator of the second fraction. The correct solution is \(\frac{x+5}{x+3}\).
1Step 1: Identify the Error
The incorrect answer has a mistake in the numerator subtraction. The second fraction in the problem is written as \(\frac{3 x+40}{x^{2}-5 x-24}\), which is incorrect. The correct subtraction should use \(\frac{5 x+40}{x^{2}-5 x-24}\).
2Step 2: Combine Like Terms
Rewrite the problem correctly with both fractions having a common denominator: \(\frac{x^{2}}{x^{2}-5x-24}-\frac{5x+40}{x^{2}-5x-24}\).
3Step 3: Subtract the Numerators
Since the denominators are the same, subtract the numerators: \(\frac{x^{2}-(5x+40)}{x^{2}-5x-24}\).
4Step 4: Simplify the Numerator
Distribute the negative sign and combine the terms in the numerator: \(\frac{x^{2}-5x-40}{x^{2}-5x-24}\).
5Step 5: Factor the Numerator and Denominator (if possible)
Factor both the numerator and the denominator to check for common factors. The numerator \(x^{2}-5x-40\) factors to \( (x-8)(x+5) \) and the denominator \(x^{2}-5x-24\) factors to \( (x-8)(x+3) \).
6Step 6: Cancel Common Factors
Cancel the common factor \(x-8\), resulting in \(\frac{x+5}{x+3}\).
Key Concepts
combining like termsfactoring polynomialscanceling common factors
combining like terms
Combining like terms is an essential step when you are simplifying algebraic fractions. This means you gather and simplify terms that have the same variable raised to the same power. For example, in the expression \(2x + 3x\), both terms are 'like' because they contain the variable \(x\). When combined, they result in \(5x\).
In the given exercise, when you combine the numerators, you are essentially performing subtraction while making sure to handle each term carefully. When we subtract one fraction from another, their denominators must be the same—a key initial step in our solution. Given the fractions \( \frac{x^2}{x^2 - 5x - 24} \) and \( \frac{5x + 40}{x^2 - 5x - 24} \), both have common denominators, allowing us to combine like terms in the numerators. This looks like:\[ \frac{x^2}{x^2 - 5x - 24} - \frac{5x + 40}{x^2 - 5x - 24}\]. Next, combine them to get one fraction \[ \frac{x^2 - (5x + 40)}{x^2 - 5x - 24} \].
Distribute the minus sign across the terms in the numerator ensuring all like terms are correctly subtracted: \[ x^2 - 5x - 40 \]. This step is crucial for maintaining algebraic integrity in your expression.
In the given exercise, when you combine the numerators, you are essentially performing subtraction while making sure to handle each term carefully. When we subtract one fraction from another, their denominators must be the same—a key initial step in our solution. Given the fractions \( \frac{x^2}{x^2 - 5x - 24} \) and \( \frac{5x + 40}{x^2 - 5x - 24} \), both have common denominators, allowing us to combine like terms in the numerators. This looks like:\[ \frac{x^2}{x^2 - 5x - 24} - \frac{5x + 40}{x^2 - 5x - 24}\]. Next, combine them to get one fraction \[ \frac{x^2 - (5x + 40)}{x^2 - 5x - 24} \].
Distribute the minus sign across the terms in the numerator ensuring all like terms are correctly subtracted: \[ x^2 - 5x - 40 \]. This step is crucial for maintaining algebraic integrity in your expression.
factoring polynomials
Factoring polynomials is a method used to express a polynomial as the product of its factors. This step is especially useful when simplifying algebraic fractions as it helps in finding and canceling common factors.
In the numerator and denominator of our fraction, we have polynomials that can be factored: \(x^2 - 5x - 40\) (numerator), and \(x^2 - 5x - 24\) (denominator).
To factor these, we look for two numbers that multiply to the constant term and add up to the coefficient of the middle term (or linear term). For the numerator \( x^2 - 5x - 40 \), we look for factors of -40 that add up to -5, which gives us \((x - 8)(x + 5)\). For the denominator \( x^2 - 5x - 24 \), the factors of -24 that add up to -5 are \((x - 8)(x + 3)\).
Writing these in factored form gives us:\[ \frac{(x - 8)(x + 5)}{(x - 8)(x + 3)} \]. This method aids in simplifying complex polynomials and is fundamental to manipulating algebraic expressions.
In the numerator and denominator of our fraction, we have polynomials that can be factored: \(x^2 - 5x - 40\) (numerator), and \(x^2 - 5x - 24\) (denominator).
To factor these, we look for two numbers that multiply to the constant term and add up to the coefficient of the middle term (or linear term). For the numerator \( x^2 - 5x - 40 \), we look for factors of -40 that add up to -5, which gives us \((x - 8)(x + 5)\). For the denominator \( x^2 - 5x - 24 \), the factors of -24 that add up to -5 are \((x - 8)(x + 3)\).
Writing these in factored form gives us:\[ \frac{(x - 8)(x + 5)}{(x - 8)(x + 3)} \]. This method aids in simplifying complex polynomials and is fundamental to manipulating algebraic expressions.
canceling common factors
Canceling common factors is the final step in simplifying algebraic fractions. It involves removing identical factors from the numerator and the denominator.
After factoring the numerator \((x - 8)(x + 5)\) and the denominator \((x - 8)(x + 3)\) of our fraction, we look for common factors to cancel out. Here, \(x - 8\) is a common factor in both the numerator and denominator.
Canceling these common factors leaves us with:\[ \frac{x + 5}{x + 3} \].
This simplified form is much easier to work with and understand. When canceling common factors, make sure each factor truly matches; otherwise, you risk simplifying incorrectly.
After factoring the numerator \((x - 8)(x + 5)\) and the denominator \((x - 8)(x + 3)\) of our fraction, we look for common factors to cancel out. Here, \(x - 8\) is a common factor in both the numerator and denominator.
Canceling these common factors leaves us with:\[ \frac{x + 5}{x + 3} \].
This simplified form is much easier to work with and understand. When canceling common factors, make sure each factor truly matches; otherwise, you risk simplifying incorrectly.
Other exercises in this chapter
Problem 92
For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistak
View solution Problem 93
$$ \text { Solve: } 800=5 k $$
View solution Problem 94
$$ \text { Solve: } 0.75=\frac{k}{60} $$
View solution Problem 94
For exercises 91-94, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistak
View solution