Problem 73 - College Algebra

Question

Find each product. $$ (7 x+5 y)^{2} $$

Step-by-Step Solution

Answer

The product of the given expression is \(49x^{2}+70xy+25y^{2}\).

Step 1: Write Down the Binomial Formula

Rewrite the problem using the binomial theorem: \((a+b)^{2}=a^{2}+2ab+b^{2}\)

Step 2: Identify a and b values

Substitute the values of \(a\) and \(b\) into the formula, where \(a = 7x\) and \(b = 5y\).

Step 3: Apply the Formula

Now, apply the formula, \((7x+5y)^{2}=(7x)^{2}+2*(7x)*(5y)+(5y)^{2}\)

Step 4: Simplify the expression

Finally, simplify the expression to get the final solution: \(49x^{2}+70xy+25y^{2}\)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols (often letters) that represent unspecified numbers or values. For example, in the expression (7x + 5y)^2, 7x and 5y are terms that are added together, where x and y are variables representing unknown quantities.

Creating algebraic expressions involves the understanding of how to combine these variables and numbers through operations such as addition, subtraction, multiplication, division, and exponentiation. An important part of working with algebraic expressions is the ability to manipulate them to simplify or solve equations. This often involves applying distributive property, combining like terms, and following the order of operations.

Binomial Formula

The binomial formula, also known as the binomial theorem for the special case of squaring a binomial, allows for the expansion of expressions raised to a power. The formula for squaring a binomial is (a+b)^2 = a^2 + 2ab + b^2 where a and b are any numbers or expressions.

For instance, when we apply the binomial formula to the expression (7x+5y)^2, we identify 7x as a and 5y as b. Following the binomial formula, the square of this binomial is expanded to (7x)^2 + 2*(7x)*(5y) + (5y)^2. The binomial formula is essential in algebra, making it much easier to handle polynomial multiplication without having to apply the distributive property multiple times.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial expression to its simplest form. This involves several steps including expanding expressions using the distributive property, combining like terms, and performing arithmetic operations. When simplifying the squared binomial (7x+5y)^2, we follow the expansion using the binomial formula, which gives us (7x)^2, 2*(7x)*(5y), and (5y)^2 as separate terms.

After this expansion, we simplify each term individually. Thus, (7x)^2 becomes 49x^2, 2*(7x)*(5y) simplifies to 70xy, and (5y)^2 simplifies to 25y^2. The final result, combining all the simplified terms, is the polynomial 49x^2 + 70xy + 25y^2. Simplifying polynomials is a fundamental skill in algebra that helps to make complex expressions more manageable and easier to evaluate or graph.