Problem 80 - College Algebra

Question

Find each product. $$ (7 x+3 y)(7 x-3 y) $$

Step-by-Step Solution

Answer

The product of the two binomial expressions \( (7x + 3y)(7x - 3y) \) is \( 49x^2 - 9y^2 \).

Step 1: Concept Understanding

Recognize that the given expression is a difference of squares expression. That's because it’s in the form of \(a + b)(a - b)\), which is a difference of squares. In this case, \(a = 7x\) and \(b = 3y\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula \(a^2 - b^2\) where \(a\) is \(7x\) and \(b\) is \(3y\). Substituting these values into the formula, we get \(a^2 - b^2 = (7x)^2 - (3y)^2\).

Step 3: Simplify the Result

Calculate the square of \(7x\) and \(3y\) individually to simplify the result. This gives \(49x^2 - 9y^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions do not include equality symbols, as they represent values rather than equations.
For example, the expression \(7x + 3y\) is made up of two terms: \(7x\) and \(3y\). Each term consists of a coefficient (a constant number) and a variable (a letter that can represent any number).

Algebraic expressions can be further simplified or manipulated using various techniques, such as factoring or expanding. They often serve as the foundation for more complex algebraic interpretations and are essential in solving equations. Having a strong understanding of algebraic expressions allows you to perform operations confidently and prepare for solving algebraic equations.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to get a new polynomial. A polynomial is an algebraic expression with more than one term, like \((7x + 3y)\).

When you multiply polynomials, you apply the distributive property, also known as FOIL (First, Outer, Inner, Last) for binomials, to ensure each term in one polynomial multiplies every term in the other polynomial. In the exercise, the multiplication of polynomials follows the pattern:

• Multiply \(7x\) by \(7x\) to get \(49x^2\)
• Multiply \(7x\) by \(-3y\) to get \(-21xy\)
• Multiply \(3y\) by \(7x\) to get \(21xy\)
• Multiply \(3y\) by \(-3y\) to get \(-9y^2\)

Afterward, similar terms can be simplified and combined, but in the difference of squares, the middle terms will always cancel out. This results in a simpler polynomial expression, confirming the importance of polynomial multiplication in algebra.

Algebraic Formulas

Algebraic formulas are key tools in algebra that provide a quick way to solve problems without performing lengthy calculations. One common algebraic formula is the difference of squares, which is used in the original exercise.

The difference of squares formula states that \[ (a + b)(a - b) = a^2 - b^2 \] This formula is particularly useful when the expression follows the pattern of one term added and another subtracted, just as in \( (7x + 3y)(7x - 3y) \). Here, you recognize the difference of squares with \(a = 7x\) and \(b = 3y\). This leads to a quick simplification:

• Square the first term: \((7x)^2 = 49x^2\)
• Square the second term: \((3y)^2 = 9y^2\)

Then, subtract to get the final expression: \[ 49x^2 - 9y^2 \] Utilizing algebraic formulas, such as the difference of squares, helps simplify complex problems and enhances problem-solving efficiency in algebra.