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Chapter 7: Problem 20

Add or subtract. Write answer in lowest terms. \(\frac{t^{2}}{t-3}+\frac{-3 t}{t-3}\)

Short Answer

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t

Step by step solution

01

- Identify the common denominator

Both fractions already have the same denominator, which is \(t-3\).
02

- Combine the numerators

Since the denominators are the same, the numerators can be combined directly: \( \frac{t^2}{t-3} + \frac{-3t}{t-3} = \frac{t^2 - 3t}{t-3} \).
03

- Factor the numerator

The numerator \( t^2 - 3t \) can be factored by taking out the common factor \( t \): \( \frac{t(t - 3)}{t-3} \).
04

- Cancel out common terms

The \( t-3 \) terms in the numerator and denominator cancel each other out: \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
When adding or subtracting fractions, finding a common denominator is essential. It allows the fractions to be combined smoothly. In this exercise, both fractions \(\frac{t^2}{t-3}\) and \(\frac{-3t}{t-3}\) have the same denominator, \(t-3\).
Having the same denominator simplifies the process, enabling us to focus on the numerators immediately.
Remember: Any fraction addition or subtraction requires equal denominators before proceeding to combine numerators.
When fractions already share a common denominator, it's like a shortcut!
combining numerators
Once it is established that fractions have a common denominator, the next step is combining the numerators. Combine them as if they were standalone terms.
For example:
\( \frac{t^2}{t-3} + \frac{-3t}{t-3} = \frac{t^2 - 3t}{t-3} \)
Notice how the denominators remain the same while the numerators are subtracted or added.
Ensure the resulting fraction correctly reflects the operation applied to the numerators.
This approach simplifies fraction operations since only the numerators need arithmetical adjustments.
factoring polynomials
After combining the numerators, we may need to simplify the expression further. Factoring polynomials involves breaking down complex expressions into simpler factors.
In this process, identify common factors in the polynomial expression.
For instance, with \(t^2 - 3t\), notice that \(t\) is a common factor:
\( t^2 - 3t = t(t-3) \)
This step makes it easier to simplify the fraction and identify if terms can cancel out.
Factoring is crucial for simplification, solving equations, and understanding polynomial relationships.
canceling terms
Finally, after factoring, look for terms that can cancel out.
Canceling terms means dividing both the numerator and denominator by the same expression, making the fraction simpler.
In our example, after factoring, we had:
\( \frac{t(t-3)}{t-3} \)
The \(t-3\) terms in both the numerator and the denominator cancel out:
\( t \)
This leaves us with a simplified result.
Canceling common terms helps reduce complex fractions to their simplest forms, making calculations and interpretations straightforward.

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Most popular questions from this chapter

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{6 s^{2}+17 s+10}{s^{2}-4} \cdot \frac{s^{2}-2 s}{6 s^{2}+29 s+20}$$

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{2}{3} \cdot \frac{5}{6} $$

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