Problem 73 - Algebra and Trigonometry

Question

In Exercises 67–82, find each product. $$ (7 x+5 y)^{2} $$

Step-by-Step Solution

Answer

The product is \( 49x^2 + 70xy + 25y^2 \).

Step 1: Identify 'a' and 'b' in the binomial

In the binomial \( (7x + 5y) ^ 2 \), the 'a' term is \( 7x \) and the 'b' term is \( 5y \).

Step 2: Apply the binomial formula

Applying the formula \( (a + b) ^ 2 = a^2 + 2ab + b^2 \) to our binomial, we get \( (7x + 5y) ^ 2 = (7x) ^ 2 + 2 * 7x * 5y + (5y) ^ 2 \).

Step 3: Simplify the expression

We then simplify the formula to obtain the solution. Simplification results in \( 49x^2 + 70xy + 25y^2 \).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a quick and efficient way of expanding binomials—that is, expressions of the form \( (a + b)^n \) where \( n \) is a non-negative integer. Instead of multiplying the binomial by itself \( n \) times, which can be cumbersome and time-consuming, the theorem offers a shortcut via a formula that expresses the result as a sum of terms involving powers of \( a \) and \( b \) and binomial coefficients.

These coefficients can be found in Pascal's Triangle or calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) where \( n \) is the power of the binomial, and \( k \) is the particular term. For example, in the expression \( (a + b)^2 \) the theorem tells us we can expand it to \( a^2 + 2ab + b^2 \) without direct multiplication. This avoids the error-prone process of manual expansion and ensures accuracy when dealing with polynomials.

Polynomial Multiplication

Polynomial multiplication involves combining two or more polynomials to create a new polynomial. When multiplying binomials, we often use methods like the FOIL (First, Outside, Inside, Last) technique, which systematizes how we distribute each term of one binomial across the terms of another. Polynomials are algebraic expressions that consist of variables raised to powers, which are whole numbers, and their respective coefficients.

For example, to multiply \( (7x + 5y)(7x + 5y) \), we would take each term in the first binomial and multiply it by each term in the second binomial, then combine like terms. This process can be made more efficient through the use of the Binomial Theorem, particularly when dealing with expressions raised to higher powers, as we saw in the example \( (7x+5y)^2 \). It's essential to apply the correct algebraic rules, such as distributing the terms properly and combining like terms, to arrive at the correct polynomial product.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them to their simplest form. This often involves combining like terms, which are terms that share the same variables raised to the same powers and performing any arithmetic operations indicated. To simplify effectively, we need to ensure that we operate within the hierarchy of operations—parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS).

For instance, after applying the Binomial Theorem to \( (7x + 5y)^2 \), we obtain a series of terms: \( 49x^2 \), \( 70xy \), and \( 25y^2 \). These terms are unlike, so we can't combine them further, and the expression \( 49x^2 + 70xy + 25y^2 \) is considered simplified. However, if we had like terms, we would add or subtract their coefficients while keeping the variable part unchanged. Simplifying helps in making expressions easier to understand and work with, especially when solving equations or working with formulas.