Factoring quadratic expressions is a fundamental skill in algebra. Understanding how to factor expressions like z² - 81 is crucial for solving equations, simplifying expressions, and progressing to more advanced mathematical concepts. This guide provides professional suggestions to help you master this skill.
Understanding the Difference of Squares
The expression z² - 81 is a classic example of a difference of squares. This specific type of quadratic expression follows a particular pattern that allows for quick and easy factoring. The general form of a difference of squares is:
a² - b² = (a + b)(a - b)
In our case, z² - 81, we can identify 'a' and 'b' as follows:
- a = z (because a² = z²)
- b = 9 (because b² = 81)
Therefore, applying the difference of squares formula, we can factor z² - 81 as:
(z + 9)(z - 9)
Step-by-Step Approach to Factoring z² - 81
To solidify your understanding, let's break down the factoring process step-by-step:
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Identify the type of expression: Recognize that z² - 81 is a binomial (two terms) and that both terms are perfect squares. This indicates a difference of squares.
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Find the square roots: Determine the square root of each term: √z² = z and √81 = 9.
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Apply the difference of squares formula: Substitute the square roots into the formula (a + b)(a - b), where a = z and b = 9. This gives you (z + 9)(z - 9).
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Verify your answer: You can always check your answer by expanding the factored form using the FOIL method (First, Outer, Inner, Last). (z + 9)(z - 9) = z² - 9z + 9z - 81 = z² - 81. This confirms that your factoring is correct.
Beyond the Basics: Expanding Your Factoring Skills
While understanding the difference of squares is essential, it's important to build upon this foundation. Here are some suggestions for expanding your factoring skills:
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Practice Regularly: The more you practice, the more comfortable and efficient you'll become at recognizing and factoring different types of quadratic expressions.
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Explore Other Factoring Techniques: Learn to factor trinomials (expressions with three terms) using methods like grouping or the AC method.
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Solve Quadratic Equations: Use your factoring skills to solve quadratic equations by setting the factored expression equal to zero and solving for the variable.
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Utilize Online Resources: Numerous online resources, including videos and interactive exercises, can provide additional support and practice opportunities.
Mastering Factoring: A Valuable Skill
Mastering the ability to factor expressions like z² - 81 is a cornerstone of algebraic proficiency. By understanding the difference of squares formula and practicing regularly, you'll build a strong foundation for more advanced mathematical concepts. Remember to break down the problem systematically, apply the correct formula, and verify your answer to ensure accuracy. With consistent effort and the right approach, factoring will become second nature.
