Table of contents
- 0. Review of Algebra4h 16m
- 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 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Factoring Polynomials
Problem 69b
Textbook Question
In Exercises 69–80, factor completely. x⁴ − 5x²y² + y⁴
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1
Recognize that the expression is a quadratic in form, where and are the variables.
Rewrite the expression as , which resembles the quadratic form .
Identify the expression as a perfect square trinomial, which can be factored as where and .
Factor the expression using the perfect square trinomial formula: .
Recognize that is a difference of squares, which can be further factored as .
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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 and solving equations. In the case of the expression x⁴ − 5x²y² + y⁴, recognizing it as a quadratic in terms of x² can facilitate the factoring process.
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Introduction to Factoring Polynomials
Difference of Squares
The difference of squares is a specific factoring technique used when an expression takes the form a² - b², which can be factored into (a - b)(a + b). In the given polynomial, recognizing patterns that resemble the difference of squares can help in breaking down the expression into simpler factors.
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Quadratic Form
A quadratic form is an expression that can be rewritten in the standard quadratic form ax² + bx + c. In this case, the expression x⁴ − 5x²y² + y⁴ can be treated as a quadratic in x², allowing us to apply techniques for factoring quadratics, such as completing the square or using the quadratic formula.
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Vertex Form
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Related Practice
Textbook Question
In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers.
x² - 4
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