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Algebraic expressions form the core building blocks of mathematical equations. They consist of variables, constants, and arithmetic operators, such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is crucial in solving equations. In the original equation, \(3(4y-3) = y(y+1)\), we see algebraic expressions on both sides.
Let's explore the first expression: \(3(4y-3)\). Here, "3" is a coefficient, and \(4y - 3\) is a binomial. The second expression, \(y(y+1)\), involves a variable raised to the first power, indicating a degree of 1 initially. Algebraic expressions like these can be expanded by using the distributive property, which is a fundamental concept for simplifying equations.

Equation simplification is the process of condensing an equation to its simplest form, making it easier to solve or analyze. In step-by-step problem-solving, expanding and simplifying equations are key parts. Let's take the equation \(12y - 9 = y^2 + y\) found after expanding the original equation.

To simplify, we rearrange the equation, moving all terms to one side:

• Move the terms \(12y\) and \(-9\) across the equality to combine like terms.
• Reorganize to obtain \(y^2 + y - 12y + 9 = 0\).

Simplification involves combining similar terms, such as the \(y\) terms in our equation, which results in \(y^2 - 11y + 9 = 0\). Simplifying like this helps us easily recognize the type of equation we're dealing with.

The degree of a polynomial tells you the highest power of the variable in the equation. It is a crucial factor in determining the nature of the polynomial. For a quadratic equation, the degree is 2.
In the final simplified form of the equation, \(y^2 - 11y + 9 = 0\), we observe that the highest power of \(y\) is 2. Hence, the equation is a quadratic equation.
Understanding polynomial degree is important as it also influences the number of solutions or roots an equation can have. For example, a quadratic equation typically has up to two roots. Recognizing the degree at this step enables us to choose appropriate methods for finding solutions, such as factoring, using the quadratic formula, or completing the square.