Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

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2:10 minutes

Problem 23

Textbook Question

Write each rational expression in lowest terms. 3(3 - t) / (t + 5)(t - 3)

Verified step by step guidance

Identify the numerator and the denominator of the rational expression: numerator is \(3(3 - t)\) and denominator is \((t + 5)(t - 3)\).

Factor the numerator if possible. Notice that \(3(3 - t)\) can be rewritten as \(-3(t - 3)\) by factoring out \(-1\).

Rewrite the expression using the factored form: \(-3(t - 3) / (t + 5)(t - 3)\).

Cancel the common factor \((t - 3)\) from the numerator and the denominator.

The expression simplifies to \(-3 / (t + 5)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. To work with rational expressions, it is essential to understand how to manipulate polynomials, including addition, subtraction, multiplication, and division. Simplifying these expressions often involves factoring polynomials to identify common factors.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original polynomial. This is crucial for simplifying rational expressions, as it allows for the cancellation of common factors in the numerator and denominator. Techniques include finding the greatest common factor (GCF) and using special products like the difference of squares.

Lowest Terms

A rational expression is said to be in lowest terms when the numerator and denominator have no common factors other than 1. To express a rational expression in lowest terms, one must factor both the numerator and denominator and cancel any common factors. This process ensures that the expression is simplified as much as possible, making it easier to work with in further calculations.