Problem 66 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$100 y^{2}-49$$

Step-by-Step Solution

Answer

The factorized form of \(100y^{2}-49\) is \((10y + 7)(10y - 7)\).

Step 1: Identify the method of factorization

Look at the given polynomial equation \(100y^{2}-49\). The given polynomial can be recognized as a difference of squares which has the form \(a^{2} - b^{2}\), which can be factored into \((a+b)(a-b)\).

Step 2: Express the equation in the form of \(a^{2} - b^{2}\)

Let's rewrite \(100y^{2}-49\) in the form of \(a^{2} - b^{2}\). So it will look like this: \((10y)^{2} - (7)^{2}\). Here, \(a = 10y\) and \(b = 7\).

Step 3: Substitute into the difference of squares formula

Now, substitute these into the difference of squares formula \((a+b)(a-b)\). It will look like this: \((10y + 7)(10y - 7)\).

Key Concepts

Difference of Squares

When faced with a polynomial expression such as \(100y^2-49\), recognizing it as a difference of squares can be immensely helpful. A difference of squares is an algebraic expression that follows the pattern \(a^2 - b^2\). This is important because such expressions can be neatly factored into two binomials: \((a+b)(a-b)\). The beauty of this lies in its simplicity and universality.
Whether you're working with simple numbers or more complex algebraic expressions, understanding that this pattern will always factor in the same way is a key insight. So when you notice an expression that looks like two squares separated by a subtraction sign, you've likely found a difference of squares. This recognition serves as a springboard to easily break down seemingly tough equations into more manageable pieces.

Algebraic Expressions

Algebraic expressions are like puzzles. They combine numbers, variables (like \(y\)), and operations (addition, subtraction, multiplication, etc.) in various intricate ways. Understanding how these pieces fit together is crucial for anyone venturing into algebra.
When working with algebraic expressions, we often aim to simplify or rearrange them. Sometimes, as with the difference of squares, we want to factor them into a product of two or more expressions. Other times, we may want to expand them or solve for a variable. The polynomial \(100y^2-49\) is a prime example of an expression that might seem daunting at first glance but becomes clearer as you dissect it using factorization techniques.

Factorization Techniques

Factorization is like a treasure hunt in mathematics, where we break down expressions into simpler, multipliable factors. There are various techniques, and mastering them opens new doors to solving complex algebraic problems.
Some popular techniques include pulling out a greatest common factor (GCF), using the difference of squares, as previously discussed, and grouping. Each technique has its ideal scenarios, like how the difference of squares is perfect for expressions like \(100y^2-49\).
Part of improving in algebra is developing an intuition for which technique to apply to a given expression, which comes with practice and a solid grasp of core concepts. These building blocks allow you to deconstruct and solve an array of polynomial expressions with confidence.