Problem 44 - Algebra A Combined Function

Question

Multiply. $$ (7 x-5)(7 x+5) $$

Step-by-Step Solution

Answer

The product is \(49x^2 - 25\).

Step 1: Identifying the Expression Type

The expression \((7x-5)(7x+5)\) is in the form \((a-b)(a+b)\), which is a difference of squares formula. This can be recognized as a pattern where \((a-b)(a+b) = a^2 - b^2\).

Step 2: Applying the Difference of Squares Formula

Using the difference of squares formula, where \(a = 7x\) and \(b = 5\), we find that \((7x-5)(7x+5) = (7x)^2 - 5^2\).

Step 3: Calculate Each Square

Calculate \((7x)^2 = 49x^2\) and \(5^2 = 25\).

Step 4: Form the Final Expression

Substitute the squares back into the difference of squares formula to get \((7x-5)(7x+5) = 49x^2 - 25\).

Key Concepts

Identifying Expressions

When working with algebraic expressions, identifying their types is similar to recognizing familiar faces in a crowd. The ability to quickly spot patterns is an invaluable skill in algebra. In our example, we have the expression \((7x-5)(7x+5)\). This setup is not random—it's actually a classic form known as the difference of squares. The formula for the difference of squares is \((a-b)(a+b) = a^2 - b^2\). This occurs when two binomials are identical except for the sign between the terms.

For instance:

• \((a-b)(a+b)\)
• \((x-y)(x+y)\)

These can easily be transformed using this recognizable pattern. Spotting these expressions is the first step towards simplifying or multiplying them efficiently.

Algebraic Multiplication

Algebraic multiplication involves multiplying terms to simplify or expand expressions. It's a foundational skill in algebra that helps translate mathematical ideas into actionable steps. Let’s look at how it applies in our example. With the expression \((7x-5)(7x+5)\), we recognize it's in the form of a difference of squares. The actual multiplication would involve applying the formula directly: \((7x)^2 - 5^2\). Here’s how each part comes together:

• Multiply each term separately: \((7x)^2\) multiplies the entire \(7x\) term with itself, and \(5^2\) multiplies 5 with itself.
• Simplify: Calculate the squares to get \(49x^2\) and \(25\).

Notice how this approach not only achieves the multiplication but also simplifies it efficiently by leveraging the difference of squares.

Polynomial Expressions

Polynomial expressions are a type of algebraic expression that includes variables raised to whole number exponents and their coefficients. They are the cornerstone of algebra, forming the basis for more advanced equations, calculus, and beyond.Let's delve into how they relate to our multiplication exercise. The expression \((7x-5)(7x+5)\) results in a polynomial after applying the difference of squares. This particular setup simplifies to a straightforward polynomial: \(49x^2 - 25\). Breaking it down further:

• Terms: Each part of the expression \(49x^2\) and \(-25\) are known as terms of the polynomial.
• Degree: The degree of the polynomial is determined by the highest exponent, which in this case is 2 from \(x^2\).
• Structure: The structure \(ax^2 + bx + c\) is common in polynomials where \(b\) would be zero here, reducing it to just \(ax^2 + c\).

Understanding polynomial expressions allows for easier manipulation and simplification in algebraic contexts. They permit mathematicians and students alike to handle more complex equations by breaking them down into understandable pieces.