Problem 74 - College Algebra

Question

Find each product. $$ (9 x+7 y)^{2} $$

Step-by-Step Solution

Answer

\((9x + 7y)^2 = 81x^2 + 126xy + 49y^2\)

Step 1: Identify \(a\) and \(b\)

The binomial is given as \(9x + 7y\). Here, \(9x\) is \(a\) and \(7y\) is \(b\).

Step 2: Apply the Square of a Binomial Formula

The formula to square a binomial is \((a + b)^2 = a^2 + 2ab + b^2\). Substituting in the values of \(a\) and \(b\) that we identified, we get \((9x + 7y)^2 = (9x)^2 + 2(9x)(7y) + (7y)^2\).

Step 3: Simplify the Expression

Simplify the expression by squaring \(9x\) and \(7y\) and multiplying out the \(2ab\) term: \((9x)^2 = 81x^2\), \(2(9x)(7y) = 126xy\), and \((7y)^2 = 49y^2\). Therefore \((9x + 7y)^2 = 81x^2 + 126xy + 49y^2\).

Key Concepts

Binomial Theorem

Understanding the Binomial Theorem is essential when working with powers of binomials. The theorem offers a systematic way to expand an expression raised to any positive integer power. It states that for any binomial \(a + b\) and a non-negative integer \(n\), the expansion is \(a + b)^n = \sum{_{k=0}^{n}} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient.

When squaring a binomial, you are essentially dealing with the case where \(n=2\). Here, the Binomial Theorem simplifies to \(a^2 + 2ab + b^2\). This pattern is consistent and makes squaring any binomial straightforward. Let's connect this to the exercise where we squared \(9x + 7y\). We applied the pattern for \(n=2\), which comes directly from the Binomial Theorem.

Polynomial Multiplication

Multiplying polynomials, including binomials, is a fundamental operation in algebra. Polynomial multiplication, such as when squaring a binomial, follows the distributive law, also known as the FOIL (First, Outer, Inner, Last) method for binomials.

In the given problem, squaring the binomial \(9x + 7y\) involves multiplying the binomial by itself. We follow the FOIL method, multiplying each term in the first binomial by each term in the second binomial and sum up all the products. The process yields the middle term \(2ab\), where \(a\) and \(b\) are the terms from the original binomial. These steps are a practical application of polynomial multiplication rules and they emphasize understanding how to systematically handle more complex expressions involving binomials.

Algebraic Expressions

The concept of algebraic expressions is the cornerstone of algebra. An algebraic expression can include constants, variables, and the operations of addition, subtraction, multiplication, division, and exponentiation by variables.

The exercise involved squaring a binomial, which itself is an algebraic expression comprised of two terms. When squaring this binomial, we end up with a trinomial as the result, which is another type of algebraic expression. The skills required in manipulating these expressions are fundamental as they apply to numerous areas in mathematics and science. The simplification of expressions, understanding how to combine like terms, and realizing the underlying structure are all part of mastering the manipulation of algebraic expressions.