Problem 92 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The expanded form of the binomial expression (x+7y)^2 is x^2 + 14xy + 49y^2.

Step 1: Identify the terms in the binomial

In the binomial (x+7y)^2, the terms are x and 7y. Now, these can be related to a and b, as required for the formula of the square of a binomial (a+b)^2 = a^2 + 2ab + b^2.

Step 2: Apply the binomial square expansion formula

Applying the formula to the binomial (x + 7y)^2 yields x^2 + 2*(x)*(7y) + (7y)^2.

Step 3: Simplify the equation

Simplifying the equation gives x^2 + 14xy + 49y^2.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this particular exercise, we are working with symbols like \(x\) and \(y\), which represent numbers but are not given a specific value. This allows us to perform operations that are valid for any numbers.

• Algebraic expressions like \((x + 7y)^2\) consist of variables (\(x\) and \(y\)) and constants.
• The operations involved include addition, multiplication, and exponentiation.

Understanding algebra is central to solving problems such as the given exercise, where we analyze and manipulate expressions using algebraic rules. In this case, we are using the power of algebra to expand and simplify a binomial expression.

Polynomial Multiplication

Polynomial multiplication is key for expanding expressions like \((x + 7y)^2\). Here, we multiply polynomials to combine several smaller expressions into one larger polynomial.In this exercise, we're turning a binomial squared into a quadratic expression. It involves multiplying each part of the binomial by each other. For example, in the expression \((x + 7y)^2\):

• The first step is to write the binomial twice as \((x + 7y)(x + 7y)\).
• You then apply the distributive property to multiply each term: \(x \cdot x\), \(x \cdot 7y\), \(7y \cdot x\), and \(7y \cdot 7y\).
• After multiplication, you combine like terms to simplify the expression.

This is essentially expanding the expression, transforming it from a compact version to an extended quadratic form.

Quadratic Expressions

A quadratic expression is a type of polynomial that involves terms up to the second degree. The standard form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.In solving \((x + 7y)^2\), the binomial square expansion results in a quadratic expression:

• The term \(x^2\) is your square of the first term.
• The term \(14xy\) comes from multiplying the two terms of the binomial and then doubling the result (from the formula \(2ab\)).
• The term \(49y^2\) is the square of the second term in the binomial.

Thus, the simplified form \(x^2 + 14xy + 49y^2\) is a quadratic expression. Here, the highest power of the variables is 2, which is why it's quadratic. Understanding how to manipulate and simplify quadratic expressions is a fundamental skill in algebra and a stepping stone to more advanced mathematical concepts.