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When simplifying rational expressions such as \( \frac{t}{t+3}+\frac{4t}{t-3}-\frac{18}{t^2-9} \), it's crucial to work with a common denominator to combine terms easily. The least common denominator (LCD) is the smallest expression that each of the original denominators can divide without a remainder.
To identify the LCD, look at each denominator: \( t+3 \), \( t-3 \), and \( t^2-9 \). Notice that \( t^2-9 \) can be rewritten using the difference of squares, which is \( (t+3)(t-3) \). Therefore, the LCD in this case is \((t+3)(t-3)\).
Using the LCD ensures that all fractions have a unified denominator, making it possible to add or subtract them by operating only on their numerators. This process simplifies the initial complex expression significantly.

Understanding the difference of squares is key in working with quadratic expressions and rational expressions. This concept is used to factor expressions like \( t^2-9 \). It follows the format \( a^2-b^2 = (a+b)(a-b) \).
For \( t^2-9 \), you can see it as \( t^2 - 3^2 \). Using difference of squares, it factors into \( (t+3)(t-3) \).
Factoring using this method makes it easier to identify the LCD or simplify the expression further. The difference of squares allows us to decompose complex terms, realize their components, and address each one individually, making algebraic manipulation more straightforward.

Once you've expressed all fractions using the least common denominator, combining them means you're ready to simplify the numerators. This involves combining like terms, which are terms in the expression that have similar variables raised to the same power.
From our combined expression: \[ t^2 - 3t + 4t^2 + 12t - 18 \]we identify like terms:

• \( t^2 \) and \( 4t^2 \) combine to \( 5t^2 \)
• \( -3t \) and \( 12t \) combine to \( 9t \)

This simplification reduces the clutter and merges similar terms, making further operations simpler, and helping achieve the simplified expression.

Factoring is the process of breaking down an expression into products of simpler expressions that can multiply together to give the original one. In the context of rational expressions, factoring helps simplify complex algebraic terms.
When looking at the numerator from the combined fractions \( 5t^2 + 9t - 18 \), factoring might further simplify the algebraic expression, although in this instance, \( 5t^2 + 9t - 18 \) does not factor neatly into simpler expressions.
However, keeping factoring in mind is still essential. Always check to see if the expression can be factored further. This could lead to canceling out terms with the denominator, sometimes helping reduce the expression to its simplest form. Always check for this potential reducing step.