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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1:52 minutes

Problem 62

Textbook Question

Factor each polynomial. See Examples 5 and 6. y^4-81

Verified step by step guidance

Recognize that the expression \( y^4 - 81 \) is a difference of squares.

Recall the formula for the difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).

Identify \( a^2 = y^4 \) and \( b^2 = 81 \), so \( a = y^2 \) and \( b = 9 \).

Apply the difference of squares formula: \( y^4 - 81 = (y^2 - 9)(y^2 + 9) \).

Notice that \( y^2 - 9 \) is also a difference of squares, so factor it further: \( y^2 - 9 = (y - 3)(y + 3) \).

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Key Concepts

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

Factoring Polynomials

Factoring polynomials 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. Common methods include factoring out the greatest common factor, using special products, and applying techniques like grouping.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions in the form a^2 - b^2, which can be factored into (a - b)(a + b). In the case of the polynomial y^4 - 81, it can be recognized as a difference of squares since y^4 is (y^2)^2 and 81 is 9^2, allowing for straightforward factoring.

Quadratic Factors

After applying the difference of squares, the resulting factors may sometimes be quadratic expressions. For instance, in the case of y^4 - 81, after factoring it as (y^2 - 9)(y^2 + 9), the factor y^2 - 9 can be further factored into (y - 3)(y + 3). Understanding how to factor quadratics is crucial for fully simplifying polynomials.