Problem 33 - Algebra A Combined Function

Question

Multiply. $$ (3 x-7 y)^{2} $$

Step-by-Step Solution

Answer

\(9x^{2} - 42xy + 49y^{2}\)

Step 1: Recognize the Formula

We need to recognize that we are asked to square a binomial expression. The binomial given is \((3x - 7y)\), and the formula for the square of a binomial \((a - b)^{2}\) is \(a^{2} - 2ab + b^{2}\). Here, \(a = 3x\) and \(b = 7y\).

Step 2: Apply the Formula

Substitute \(a = 3x\) and \(b = 7y\) into the formula. The expression becomes:\[(3x - 7y)^{2} = (3x)^{2} - 2 \, (3x)(7y) + (7y)^{2}\]

Step 3: Calculate Each Term

Now calculate each part of the expression:- \((3x)^{2} = 9x^{2}\)- \(-2 \, (3x)(7y) = -42xy\)- \((7y)^{2} = 49y^{2}\)

Step 4: Combine the Terms

Combine the individual terms calculated in Step 3 to get:\[9x^{2} - 42xy + 49y^{2}\]

Step 5: Write the Final Answer

The multiplication of the given binomial squared results in:\[(3x - 7y)^{2} = 9x^{2} - 42xy + 49y^{2}\]

Key Concepts

Squaring a Binomial

Squaring a binomial involves taking a binomial expression and multiplying it by itself. A binomial is an algebraic expression that consists of two terms. The process can initially seem complicated, but it becomes quite straightforward once you understand the fundamental pattern used in the binomial square formula.

When we talk about \(a - b\) as a binomial being squared, it means we directly use the expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here's a breakdown of each part of this formula:

• \(a^2\) represents the square of the first term.
• -2ab accounts for two times the product of the two terms.
• \(b^2\) is the square of the second term.

The formula simplifies the multiplication process and ensures all components of the calculation are accounted for. This pattern applies whether the signs between the terms are positive or negative; however, the middle term will change accordingly.

Polynomial Multiplication

Polynomial multiplication is the process through which one polynomial is multiplied by another. When multiplying polynomials, each term of one polynomial must multiply every term of the other polynomial. This ensures that every possible combination is accounted for in the product.

Specifically, when squaring a binomial, as shown in square of a binomial, the terms are multiplied systematically, following the distributive property. Here’s how it breaks down:

• First, multiply the first term of the binomial by itself, giving us \(a^2\).
• Next, multiply the first term of the first binomial by the second term of the second binomial, and the same goes with the second term of the binomial, applied with \(-2\).
• Finally, multiply the second term of the binomial by itself to give \(b^2\).

The expanded result is then combined to form a simpler expression by adding or subtracting the terms if possible. This mechanism is crucial for correctly expanding binomials and simplifying larger polynomials.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations like addition and multiplication, arranged in a mathematical phrase. They are used to represent real-world situations using letters and numbers. In this context, expressions such as \((3x - 7y)^2\) are an example since they contain variables and coefficients grouped using mathematical operations.

When we engage in operations like squaring a binomial or polynomial multiplication, we manipulate these expressions to simplify or modify them. Algebraic expressions need to be handled with precision:

• The coefficients (numerical values in expressions like 3 in 3x) and the variables need careful attention to ensure terms are correctly generated.
• The use of proper mathematical laws and formulas, like the binomial theorem, ensures accuracy in results.

Mastering the use of algebraic expressions is a core skill in mathematics that sets the foundation for solving equations and modelling mathematical scenarios in advanced math. They provide a clear and structured way for representing and calculating mathematical problems.