Problem 23 - Elementary and Intermediate Algebra

Question

Factor difference of two squares. \(16 t^{2}-25 w^{4}\)

Step-by-Step Solution

Answer

The factorization is \((4t - 5w^2)(4t + 5w^2)\).

Step 1: Identify the difference of squares

The expression given is \(16t^2 - 25w^4\). Recognize that this is a difference of two squares since both \(16t^2\) and \(25w^4\) are perfect squares.

Step 2: Express each term as a square

Write each term as a square. We have \( (4t)^2 = 16t^2 \) and \( (5w^2)^2 = 25w^4 \). Thus, we can rewrite the expression as \((4t)^2 - (5w^2)^2\).

Step 3: Apply the difference of squares formula

The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). Here, let \(a = 4t\) and \(b = 5w^2\). Apply the formula to get \((4t - 5w^2)(4t + 5w^2)\).

Step 4: Verify the factoring

Multiply the factors to verify the solution: \((4t - 5w^2)(4t + 5w^2) = (4t)^2 - (5w^2)^2 = 16t^2 - 25w^4\). The original expression is regained, confirming the factorization is correct.

Key Concepts

Factoring Techniques

Factoring techniques are vital tools in algebra that help simplify expressions and solve equations. Factoring is the process of breaking down an expression into simpler pieces that, when multiplied together, give back the original expression. There are several methods used in factoring such as:

• Factoring out the greatest common factor
• Factoring trinomials
• Using special products like the difference of squares

For the difference of squares, the key is to recognize the pattern: a square minus another square. This means identifying two perfect squares separated by a subtraction sign. Understanding this process can significantly simplify problems and make algebra much more manageable. Applying these techniques efficiently requires practice and familiarity with various algebraic identities.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. In expressions like \(16t^2 - 25w^4\), understanding the components is crucial.

• Numbers like 16 and 25 are coefficients.
• Variables such as \(t\) and \(w\) represent unknown values.
• Operations include addition, subtraction, multiplication, or division.

The power of algebra is in its ability to express complex relationships with these symbols. Unlike numerical expressions, algebraic expressions can be simplified or manipulated using various methods, including factoring. When working with them, pay close attention to the properties of exponents and the order of operations to ensure accuracy.

Perfect Squares

Perfect squares play a pivotal role in many algebraic processes, especially when working with factoring. A perfect square is simply the product of an integer multiplied by itself, such as \((4t)^2\) or \((5w^2)^2\). Recognizing perfect squares can help identify opportunities to apply the difference of squares formula.Here are some tips for identifying perfect squares:

• If a number can be expressed as a whole number squared, it is a perfect square (e.g., 16 is \(4^2\)).
• A term like \(x^2\) is inherently a perfect square.
• When dealing with variables, ensure that the exponent is even to confirm that it can be a perfect square.

By understanding and recognizing perfect squares, you open up easier pathways to solve algebraic problems efficiently.