Factored Form Of A²-121 How To Solve Difference Of Squares

Factoring quadratic expressions is a fundamental skill in algebra, and among the various factoring techniques, the difference of squares pattern holds a prominent place. It's a powerful tool that allows us to simplify expressions and solve equations with ease. In this article, we will delve into the heart of this pattern and demonstrate how to apply it to factor the expression a² - 121. Understanding and mastering this technique will significantly enhance your algebraic abilities and pave the way for tackling more complex mathematical problems.

Understanding the Difference of Squares Pattern

The difference of squares pattern is a special case of factoring that arises when we have an expression in the form of A² - B². This pattern states that the difference of two perfect squares can be factored into the product of two binomials: (A - B)(A + B). This elegant pattern stems from the distributive property of multiplication, also known as the FOIL (First, Outer, Inner, Last) method, when expanding the product of the two binomials. Let's explore how this works:

(A - B)(A + B) = A(A + B) - B(A + B)

= A² + AB - BA - B²

= A² - B² (Since AB and BA are the same, they cancel each other out)

This concise derivation showcases the beauty and symmetry inherent in the difference of squares pattern. The middle terms conveniently cancel out, leaving us with the difference of the squares of A and B. Recognizing this pattern is crucial for efficient factoring. To effectively apply the difference of squares pattern, you must be able to identify perfect squares and understand how they relate to their square roots. A perfect square is a number or expression that can be obtained by squaring another number or expression. For example, 9 is a perfect square because it is the result of squaring 3 (3² = 9). Similarly, x² is a perfect square because it is the result of squaring x. Recognizing perfect squares quickly is essential for applying the difference of squares pattern. Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. These numbers are the squares of the integers from 1 to 12, and becoming familiar with them will significantly speed up your factoring process.

Applying the Pattern to a² - 121

Now, let's apply this pattern to the specific expression we are interested in: a² - 121. The first step is to recognize whether the expression fits the form of A² - B². We can see that a² is clearly a perfect square, as it is the square of 'a'. The next crucial step is to determine if 121 is also a perfect square. Recall that a perfect square is a number that can be obtained by squaring an integer. Is 121 the square of an integer? To find out, we need to determine the square root of 121. The square root of 121 is 11 because 11 multiplied by itself (11²) equals 121. Therefore, 121 is indeed a perfect square. Now that we have identified both terms as perfect squares, we can express the expression as a difference of squares:

a² - 121 = a² - 11²

This representation makes it clear that we can apply the difference of squares pattern. In this case, A corresponds to 'a' and B corresponds to 11. Now, we can directly apply the pattern:

A² - B² = (A - B)(A + B)

Substituting 'a' for A and 11 for B, we get:

a² - 11² = (a - 11)(a + 11)

Therefore, the factored form of a² - 121 is (a - 11)(a + 11). This result demonstrates the power and efficiency of the difference of squares pattern. By recognizing the pattern and applying the formula, we can quickly and easily factor expressions that might otherwise seem daunting.

Why the Other Options Are Incorrect

To solidify our understanding, let's analyze why the other options provided are incorrect. This will help us further grasp the difference of squares pattern and avoid common mistakes.

Option A: (a - 121)(a + 1)

This option is incorrect because it doesn't correctly identify the square root of 121. It seems to treat 121 as if it were the square root, rather than the square of a number. When we expand this expression using the FOIL method, we get:

(a - 121)(a + 1) = a² + a - 121a - 121

= a² - 120a - 121

This result is not equal to the original expression, a² - 121, indicating that this factorization is incorrect. The key error here is not recognizing that 121 is 11 squared and applying the difference of squares pattern correctly.

Option C: (a + 11)(a + 11)

This option represents the square of a binomial, specifically (a + 11)². While it does involve 11, it doesn't represent the difference of squares. Expanding this expression, we get:

(a + 11)(a + 11) = a² + 11a + 11a + 121

= a² + 22a + 121

Again, this result does not match the original expression, a² - 121. This option incorrectly assumes that the expression is a perfect square trinomial, rather than a difference of squares. The presence of the middle term, 22a, clearly indicates that this is not the correct factorization.

Option D: (a - 121)(a - 1)

Similar to option A, this option misidentifies the square root of 121 and incorrectly applies the pattern. Expanding this expression gives us:

(a - 121)(a - 1) = a² - a - 121a + 121

= a² - 122a + 121

This result also does not equal the original expression, a² - 121. The error lies in not recognizing the difference of squares pattern and misinterpreting the role of 121 in the expression. These incorrect options highlight the importance of carefully identifying perfect squares and correctly applying the difference of squares pattern. Understanding why these options are wrong reinforces the correct method and helps prevent similar errors in the future.

Conclusion: Mastering the Difference of Squares

In conclusion, the factored form of a² - 121 is (a - 11)(a + 11). This factorization is a direct application of the difference of squares pattern, which states that A² - B² = (A - B)(A + B). By recognizing that a² is the square of 'a' and 121 is the square of 11, we can easily apply this pattern to obtain the correct factored form.

Mastering the difference of squares pattern is a crucial step in developing your algebraic skills. It's a technique that appears frequently in various mathematical contexts, from simplifying expressions to solving equations. The ability to quickly identify and apply this pattern will save you time and effort, and it will also provide a solid foundation for tackling more advanced factoring problems.

Remember, the key to success with the difference of squares lies in recognizing perfect squares and understanding the underlying pattern. Practice with various examples, and you'll soon become proficient in applying this powerful factoring technique. Keep exploring and expanding your mathematical knowledge, and you'll be amazed at the connections and patterns you discover!