Problem 69 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (3 x+5 y)(3 x-5 y) $$

Step-by-Step Solution

Answer

The product of \((3x+5y)(3x-5y)\) is \(9x^2 - 25y^2\).

Step 1: Identity of the Difference of Squares formula

Recognize that the given expression is in the form of a Difference of Squares formula. This is denoted as \(a+b)(a-b) = a^2 - b^2\), where a is \(3x\) and b is \(5y\).

Step 2: Apply the Difference of Squares formula

Apply the Difference of Squares formula by squaring the first term and subtracting the square of the second term. This results in \((3x)^2 - (5y)^2\).

Step 3: Calculate the squares

Calculate the squares of the two terms. \((3x)^2\) gives \(9x^2\), and \((5y)^2\) gives \(25y^2\).

Step 4: Write down the final expression

Subtract the square of the second term from that of the first term, giving the final result of \(9x^2 - 25y^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like x or y), and operation symbols. They are the building blocks of algebra and are used to describe patterns, relationships, and changes. The expression \(3x+5y\) represents a sum of two terms: \(3x\), which is three times the variable x, and \(5y\), which is five times the variable y. When you encounter algebraic expressions, it's essential to recognize their form and understand how to manipulate them, as this can simplify complex problems into more manageable ones.

For example, in the textbook exercise provided, the expression \(3x+5y\) is paired with \(3x-5y\), creating a scenario that hints at a special product known as the difference of squares. Recognizing forms like these are key to mastering polynomial multiplication and algebra at large.

Squaring Binomials

Squaring a binomial means multiplying the binomial by itself, which is a special case of polynomial multiplication. A binomial is an algebraic expression with two terms, for example, \(a+b\) or \(a-b\). When you square a binomial, you apply the formula \(a+b)^2 = a^2 + 2ab + b^2\). However, the exercise we're looking at involves the difference of two squares, which is a related, but different, concept.

When you have a difference of two squares, such as \(a+b)(a-b)\), the middle terms cancel out, leaving you with just \(a^2-b^2\). It's a shortcut that simplifies calculations and is very useful in algebra. Understanding how to square binomials helps students recognize patterns like the difference of squares, leading to quicker and more accurate problem-solving.

Polynomial Multiplication

Polynomial multiplication involves multiplying polynomials together to get a new polynomial. A polynomial is an expression made up of variables raised to whole number exponents and their coefficients. When multiplying polynomials, each term in the first polynomial is multiplied by every term in the second polynomial using the distributive property—often visualized through methods such as the FOIL (First, Outer, Inner, Last) method for binomials.

The exercise we're discussing uses polynomial multiplication in the context of the difference of squares. The students can use this special case to streamline the multiplication process, focusing on squaring the individual terms, because due to the symmetrical nature of the binomials in question, the cross-terms cancel each other out. This concept is vital as it allows for simplification when multiplying more complex polynomial expressions.