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The binomial squaring formula is a powerful tool that helps us expand expressions of the form \((a+b)^2\). This formula states that \((a+b)^2 = a^2 + 2ab + b^2\), which allows us to break down the square of a binomial into simpler, more manageable terms. In our specific example, \((3b+7)^2\), we identify \(a\) as \(3b\) and \(b\) as \(7\). Using the formula, we expand the expression to:

• Square the first term: \((3b)^2 = 9b^2\).
• Multiply the two terms together and double the product: \(2(3b)(7) = 42b\).
• Square the second term: \((7)^2 = 49\).

This expands the original expression into \(9b^2 + 42b + 49\). This step-by-step breakdown makes it simpler to understand how each part of the formula logically transforms into the final expanded result. When dealing with the squaring of binomials, this formula is essential and frequently used.

Polynomial expressions consist of variables and coefficients that are combined using only addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is crucial when working with algebraic expressions because of their frequent appearance in mathematical problems. When expanding a binomial like \((3b+7)^2\), we're essentially forming a polynomial expression by applying the binomial squaring formula. The terms \(9b^2\), \(42b\), and \(49\) are the result of expanding the original binomial.

• The term \(9b^2\) is a quadratic term indicating the presence of \(b^2\), and \(9\) is its coefficient.
• The term \(42b\) is a linear term, with \(42\) as its coefficient.
• The last term \(49\) is a constant term, having no variable attached.

Each part of these expressions plays a role in defining the polynomial's behavior and characteristics, such as its degree, leading coefficient, and constant term.

Once a polynomial expression is formed, simplification is key to making calculations more manageable and solutions more understandable. In our exercise, after applying the binomial squaring formula to \((3b+7)^2\), we obtained \(9b^2 + 42b + 49\). However, the expression was part of an entire equation, \(2(3b+7)^2\). This leads us to algebraic simplification. To simplify further:

• Multiply each term inside the parentheses by the outer coefficient, which is \(2\).
• This results in: \(2(9b^2) + 2(42b) + 2(49) = 18b^2 + 84b + 98\).

Simplification not only helps the expression become easier to handle but also aids in the clarity of the solution. The simplified polynomial \(18b^2 + 84b + 98\) is easier to interpret, and any further comparisons, evaluations, or operations become straightforward. Learning how to simplify expressions efficiently is invaluable while working with more complex algebraic problems.