Problem 6
Question
Find all numbers that must be excluded from the domain of each rational expression. $$ \frac{x-3}{x^{2}+4 x-45} $$
Step-by-Step Solution
Verified Answer
The values that must to be excluded from the domain of the given rational expression are \(x = -9\) and \(x = 5\).
1Step 1: Identify the denominator of the rational expression
The denominator of our rational expression is \(x^{2} + 4x - 45\). We need to find the values for \(x\) which make this denominator zero.
2Step 2: Set the denominator equal to zero and solve for \(x\)
So, we get the equation as \(x^{2} + 4x - 45 = 0\). Solve this equation for \(x\). We can factorize the equation as \((x+9)(x-5) = 0\). This yields us two solutions, \(x = -9\) and \(x = 5\).
3Step 3: Exclude the solutions obtained from the domain
The solutions obtained are the values for which denominator becomes zero. Hence, these values must be excluded from the domain of the expression.
Key Concepts
Excluded ValuesSolving EquationsFactoring Quadratics
Excluded Values
Understanding excluded values in the context of rational expressions is crucial for a clear grasp of algebra. When working with a rational expression, the denominator dictates which values can be used for the variable in question. By definition, a rational expression cannot have a denominator of zero, as division by zero is undefined in mathematics.
To identify excluded values, one simply needs to look at the denominator of the expression and set it equal to zero, then solve for the variable. These solutions are the values that must be excluded from the domain. For example, given the rational expression \(\frac{x-3}{x^{2}+4 x-45}\), the denominator is \(x^2 + 4x - 45\). Setting the denominator equal to zero gives us the equation \(x^2 + 4x - 45 = 0\), which, when solved, excludes \(x = -9\) and \(x = 5\) from the domain, ensuring the expression remains valid for all other real numbers.
To identify excluded values, one simply needs to look at the denominator of the expression and set it equal to zero, then solve for the variable. These solutions are the values that must be excluded from the domain. For example, given the rational expression \(\frac{x-3}{x^{2}+4 x-45}\), the denominator is \(x^2 + 4x - 45\). Setting the denominator equal to zero gives us the equation \(x^2 + 4x - 45 = 0\), which, when solved, excludes \(x = -9\) and \(x = 5\) from the domain, ensuring the expression remains valid for all other real numbers.
Solving Equations
The process of solving equations, specifically in regard to rational expressions, entails finding the value of the variable that satisfies the equation. When factoring quadratic equations, as seen in the step-by-step solution, the goal is to break down a quadratic formula into products of binomials. Here, \(x^{2} + 4x - 45 = 0\) is factorized as \( (x+9)(x-5) = 0 \).
Next, application of the Zero Product Property informs us that if the product of two factors is zero, then at least one of the factors must be zero. Thus, we can solve the binomials separately: either \(x + 9 = 0\), which yields \(x = -9\), or \(x - 5 = 0\), leading to \(x = 5\). These values cannot be included in the domain of the original expression because they would make the denominator zero, resulting in an undefined expression.
Next, application of the Zero Product Property informs us that if the product of two factors is zero, then at least one of the factors must be zero. Thus, we can solve the binomials separately: either \(x + 9 = 0\), which yields \(x = -9\), or \(x - 5 = 0\), leading to \(x = 5\). These values cannot be included in the domain of the original expression because they would make the denominator zero, resulting in an undefined expression.
Factoring Quadratics
The skill of factoring quadratics is a cornerstone in algebra that significantly simplifies solving quadratic equations. Factoring involves rewriting a quadratic expression as a product of two linear expressions (binomials). To successfully factor a quadratic expression like \( x^{2} + 4x - 45 \), one should find two numbers that both add to the middle term's coefficient (in this case, 4) and multiply to the constant term (in this case, -45).
In our example, these numbers are 9 and -5. Thus, the quadratic expression factors to \( (x+9)(x-5) \). This is a quick way to solve for the roots of the quadratic equation, leading to the aforementioned excluded values in the domain of the rational expression. Developing fluency in this technique eases the path to uncovering the domain constraints of rational expressions, ensuring students have the tools to navigate algebraic challenges.
In our example, these numbers are 9 and -5. Thus, the quadratic expression factors to \( (x+9)(x-5) \). This is a quick way to solve for the roots of the quadratic equation, leading to the aforementioned excluded values in the domain of the rational expression. Developing fluency in this technique eases the path to uncovering the domain constraints of rational expressions, ensuring students have the tools to navigate algebraic challenges.
Other exercises in this chapter
Problem 5
Evaluate each exponential expression. $$ -2^{6} $$
View solution Problem 5
$$\text {find the degree of the polynomial.}$$ $$3 x^{2}-5 x+4$$
View solution Problem 6
In Exercises \(1-10\), factor out the greatest common factor. $$6 x^{4}-18 x^{3}+12 x^{2}$$
View solution Problem 6
Evaluate each expression in Exercises \(1-6\) or indicate that the root is not a real number. $$\sqrt{(-17)^{2}}$$
View solution