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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1:58 minutes

Problem 135

Textbook Question

Factor each polynomial over the set of rational number coefficients. (25/9)x^4-(9y^2)

Verified step by step guidance

Step 1: Identify the polynomial to be factored: \( \frac{25}{9}x^4 - 9y^2 \).

Step 2: Recognize that this expression is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).

Step 3: Rewrite each term as a square: \( \left(\frac{5}{3}x^2\right)^2 - (3y)^2 \).

Step 4: Apply the difference of squares formula: \( \left(\frac{5}{3}x^2 - 3y\right)\left(\frac{5}{3}x^2 + 3y\right) \).

Step 5: Verify that the factors are correct by expanding them back to the original polynomial.

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Key Concepts

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

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions, solving equations, and analyzing polynomial behavior. Understanding how to identify common factors, apply the difference of squares, or use techniques like grouping is crucial for effective factoring.

Rational Coefficients

Rational coefficients are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In the context of polynomial factoring, it is important to ensure that all coefficients in the factored form remain rational. This concept is vital for maintaining the integrity of the polynomial when performing operations such as factoring or simplifying.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It provides insight into the polynomial's behavior, including the number of roots and the shape of its graph. In the given polynomial, recognizing the degree helps in determining the appropriate factoring techniques and understanding the overall structure of the polynomial.