Adding and Subtracting Rational Expressions With Unlike Denominators

Summary

This lesson focuses on adding and subtracting rational expressions with unlike denominators. It demonstrates step-by-step methods for finding common denominators and combining fractions through various examples. Key concepts include factoring polynomials, distributing terms, and simplifying expressions. The lesson emphasizes the importance of manipulating expressions correctly to arrive at the final simplified form, ensuring clarity in each operation. Through multiple examples, learners gain a deeper understanding of the process, ultimately culminating in a clear presentation of final answers in factored forms.

Takeaways

• 😀 Understanding rational expressions with unlike denominators is crucial for combining fractions.
• 😀 To add or subtract fractions, first find a common denominator using the least common multiple.
• 😀 When adjusting fractions, multiply the numerator and denominator by necessary factors to achieve a common denominator.
• 😀 Combining the numerators is straightforward once the fractions share a common denominator.
• 😀 Simplification of the resulting expression may involve factoring and combining like terms.
• 😀 Distributing negative signs correctly when adjusting fractions is essential to maintain accuracy.
• 😀 Factoring expressions can simplify the process and help in identifying common factors.
• 😀 Always check if the final result can be factored further for simplification.
• 😀 Be cautious with negative signs during distribution to avoid errors in the final expression.
• 😀 The final answer should be expressed in its simplest form, ensuring all possible simplifications are made.

Q & A

What is the first step when adding or subtracting rational expressions with unlike denominators?

The first step is to find a common denominator for the fractions.

How do you determine the least common multiple (LCM) of two denominators?

You can find the LCM by identifying the smallest multiple that both denominators share. For example, for 4 and 3, the LCM is 12.

In the example of (5/x) + (3/x^2), how do you combine these fractions?

You multiply the first fraction by x/x to get a common denominator of x^2, resulting in (5x + 3)/x^2.

What must you do when you encounter a negative sign in front of a fraction?

You need to distribute the negative sign across the numerator when combining fractions.

When combining the fractions (4/(x-2)) + (5/(x+2)), what is the common denominator?

The common denominator is (x-2)(x+2).

How do you factor the expression x^2 - x - 20?

You can factor it as (x - 5)(x + 4) since -5 and +4 multiply to -20 and add to -1.

What is the purpose of factoring out a negative one from a denominator?

Factoring out a negative one helps to rewrite the denominator in a form that matches other fractions for easier combination.

What steps do you follow to combine multiple fractions into a single fraction?

Multiply each fraction by the necessary factors to achieve a common denominator, then combine the numerators over that common denominator.

In the example of adding (5/(x+2)) + (2/(x+1)) - (3/(x-1)), what is the first calculation?

You first find a common denominator, which in this case is (x+2)(x+1)(x-1), and then distribute each term accordingly.

What is the final answer when combining and simplifying the fractions in the last example?

The final answer, in its complete factored form, is (4x + 5)(x - 3) divided by the common denominator.