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 78c
Textbook Question
Factor: x² + 4x + 4 − 9y². (Section 5.6, Example 4)
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Step 1: Identify the structure of the expression. Notice that the expression is a combination of a quadratic trinomial and a difference of squares.
Step 2: Focus on the quadratic trinomial part, . Recognize it as a perfect square trinomial.
Step 3: Factor the perfect square trinomial as .
Step 4: Recognize the expression as a difference of squares.
Step 5: Apply the difference of squares formula, , to factor into .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting them as a product of two binomials. In the expression x² + 4x + 4, we can recognize it as a perfect square trinomial, which factors to (x + 2)². Understanding how to identify and factor these forms is essential for simplifying expressions and solving equations.
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Difference of Squares
The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). In the given expression, after factoring the quadratic part, we can recognize that we have a difference of squares when we rewrite it as (x + 2)² - (3y)². This allows us to apply the difference of squares formula to further factor the expression.
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Binomial Products
Binomial products are expressions formed by multiplying two binomials. When we apply the difference of squares to (x + 2)² - (3y)², we obtain (x + 2 - 3y)(x + 2 + 3y). Understanding how to expand and simplify binomial products is crucial for both factoring and solving polynomial equations.
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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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