Problem 9
Question
Add or Subtract the following rational expressions. $$ \frac{8 x^{2}+x-1}{x^{2}-6 x+8}+\frac{2 x^{2}+3 x}{x^{2}-6 x+8}-\frac{5 x^{2}+3 x-4}{(x-4)(x-2)} $$
Step-by-Step Solution
Verified Answer
Question: Add and subtract the following rational expressions:
$$
\frac{8x^2 + x -1}{x^2 - 6x + 8}, \frac{2x^2 +3x}{x^2 - 6x + 8}, \text{ and } \frac{5x^2 + 3x -4}{(x-4)(x-2)}.
$$
Answer: After adding and subtracting the given rational expressions, we get the simplified result:
$$
\frac{5x^2 - x + 3}{x^2 - 6x + 8}
$$
1Step 1: Find the Common Denominator
All three expressions have the same denominator, \((x^2 - 6x + 8)\), except the third expression has a different denominator - \((x-4)(x-2)\). Since the other two expressions have the same denominator, we can conclude that the third denominator is already the common denominator for all three rational expressions. So, our common denominator will be \((x-4)(x-2)\) or \((x^2 - 6x + 8)\).
2Step 2: Simplify the Third Rational Expression
The third expression has the denominator \((x-4)(x-2)\), which is not yet simplified. Simplify it and rewrite the expression as:
$$
\frac{5x^2 + 3x - 4}{(x-4)(x-2)} = \frac{5x^2 + 3x - 4}{x^2 - 6x + 8}
$$
3Step 3: Combine the Expressions Numerators
Now that all the denominators are the same, we can directly add and subtract the numerators:
$$
\frac{8x^2 + x - 1}{x^2 - 6x + 8} + \frac{2x^2 + 3x}{x^2 - 6x + 8} - \frac{5x^2 + 3x - 4}{x^2 - 6x + 8}
$$
Combine the numerators and place them over the common denominator:
$$
\frac{(8x^2 + x - 1) + (2x^2 + 3x) - (5x^2 + 3x - 4)}{x^2 - 6x + 8}
$$
4Step 4: Simplify the Resulting Expression
Perform the addition and subtraction in the numerator, and simplify the expression:
$$
\frac{8x^2 + x - 1 + 2x^2 + 3x - 5x^2 - 3x + 4}{x^2 - 6x + 8} = \frac{5x^2 - x + 3}{x^2 - 6x + 8}
$$
The expression is now simplified, and the final result is:
$$
\frac{5x^2 - x + 3}{x^2 - 6x + 8}
$$
Key Concepts
Common DenominatorAlgebraic FractionsPolynomial Simplification
Common Denominator
When working with rational expressions, particularly when adding or subtracting them, it's essential to find a common denominator. This concept is similar to finding a common ground for fractions to stand on—it’s about ensuring that all terms speak the same 'denominational' language before they can be combined. A common denominator represents a shared multiple of the original denominators which allows for the direct sum or difference of the numerators.
In our exercise, we identified that \(x^2 - 6x + 8\)\ and \(x-4)(x-2)\) are essentially the same polynomial, just represented differently. The factored form and the expanded form of a polynomial can often both serve as a common denominator. Ensuring they are in a matching format is critical for simplifying the expressions accurately and effectively. The efficiency of the process hinges on recognizing these equivalent forms.
Common mistakes often involve overlooking the potential to factor denominators or to recognize equivalent expressions. Students should always be ready to factor or expand polynomials to uncover the true common denominator.
In our exercise, we identified that \(x^2 - 6x + 8\)\ and \(x-4)(x-2)\) are essentially the same polynomial, just represented differently. The factored form and the expanded form of a polynomial can often both serve as a common denominator. Ensuring they are in a matching format is critical for simplifying the expressions accurately and effectively. The efficiency of the process hinges on recognizing these equivalent forms.
Common mistakes often involve overlooking the potential to factor denominators or to recognize equivalent expressions. Students should always be ready to factor or expand polynomials to uncover the true common denominator.
Algebraic Fractions
Algebraic fractions are simply fractions where the numerator, the denominator, or both, are algebraic expressions—usually polynomials. Simplifying algebraic fractions is an integral part of algebra that allows us to decrease the complexity of expressions and solve equations more efficiently.
When we talk about algebraic fractions, particularly in the context of summing or subtracting them as we do in this exercise, we must recognize the importance of the aforementioned common denominator. Only when the denominators are the same across all algebraic fractions can we proceed to combine the numerators. This concept adheres to the basic principles of fractions that we learn with numbers; it just incorporates variables into the mix.
To gain confidence with algebraic fractions, practice is key, especially with sums and differences that necessitate finding a common denominator to begin with. Students should familiarize themselves with factoring and simplifying polynomials, as these skills are crucial when dealing with rational expressions.
When we talk about algebraic fractions, particularly in the context of summing or subtracting them as we do in this exercise, we must recognize the importance of the aforementioned common denominator. Only when the denominators are the same across all algebraic fractions can we proceed to combine the numerators. This concept adheres to the basic principles of fractions that we learn with numbers; it just incorporates variables into the mix.
To gain confidence with algebraic fractions, practice is key, especially with sums and differences that necessitate finding a common denominator to begin with. Students should familiarize themselves with factoring and simplifying polynomials, as these skills are crucial when dealing with rational expressions.
Polynomial Simplification
Polynomials are expressions consisting of variables and coefficients, involving terms that are added, subtracted, or multiplied together (but not divided). Polynomial simplification is the process of reducing the complexity of these expressions. Simplifying a polynomial may involve combining like terms, factoring, expanding, or even dividing by common factors.
In our particular problem, once we established the common denominator, the next step was to combine the numerators to simplify the polynomial. This is done by performing arithmetic on like terms—terms that have the same variable to the same power. For example, \(8x^2\) and \(2x^2\) are like terms and can be added together. The goal here is to create the simplest expression that represents the same quantity.
The final step is often to check if the resulting polynomial can be further factored or simplified. It's not unusual to find that a polynomial simplification leads to a form that reveals more about the nature of the equation or function at hand. In a classroom or homework scenario, this understanding enables students to approach more complex algebraic problems with greater ease and intuition.
In our particular problem, once we established the common denominator, the next step was to combine the numerators to simplify the polynomial. This is done by performing arithmetic on like terms—terms that have the same variable to the same power. For example, \(8x^2\) and \(2x^2\) are like terms and can be added together. The goal here is to create the simplest expression that represents the same quantity.
The final step is often to check if the resulting polynomial can be further factored or simplified. It's not unusual to find that a polynomial simplification leads to a form that reveals more about the nature of the equation or function at hand. In a classroom or homework scenario, this understanding enables students to approach more complex algebraic problems with greater ease and intuition.
Other exercises in this chapter
Problem 9
For the following problems, solve the rational equations. $$ \frac{x}{28}=\frac{3}{7} $$
View solution Problem 9
For the following problems, perform the multiplications and divisions. $$ \frac{4 a^{3}}{5 b} \cdot \frac{3 b}{2 a} $$
View solution Problem 9
For the following problems, reduce each rational expression to lowest terms. $$ \frac{8}{4 a-16} $$
View solution Problem 10
$$ \frac{18 x^{3} y}{5 a^{2}} \cdot \frac{15 a^{3} b}{6 x^{2} y} $$
View solution