Question

$$\left(18 y^{2}-12 x y\right)-\left(6 y^{2}+x y-a\right)$$

Short answer

\(12 y^2 - 13 x y + a\)

Step-by-step solution

Distribute the negative sign

Apply the distributive property of multiplication over subtraction to distribute the negative sign to each term in the second pair of parentheses: \( (18 y^{2} - 12 x y) - (6 y^{2} + x y - a) = 18 y^{2} - 12 x y - 6 y^{2} - x y + a \)

Combine like terms

Combine the terms that involve the same variables to the corresponding powers: \((18 y^{2} - 6 y^{2}) - (12 x y + x y) + a = 12 y^{2} - 13 x y + a \)

Write the simplified expression

After combining like terms, the simplified expression is: \[12 y^{2} - 13 x y + a\]

Key concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions and solve equations efficiently. When you have a subtraction problem involving two sets of parentheses, like in the exercise \( (18 y^{2}-12 x y)-(6 y^{2}+x y-a) \), the distributive property enables you to 'distribute' the subtraction sign to each term inside the second parentheses.

This is similar to how you would distribute a multiplication factor across terms inside parentheses. The process transforms the expression into \( 18 y^{2} - 12 x y - 6 y^{2} - x y + a \), effectively removing the parentheses and changing the signs of the terms that were inside the second set.

Combining Like Terms

Once the distributive property is applied, the next step is combining like terms to simplify the algebraic expression. Like terms are terms in an expression that have exactly the same variable components raised to the same powers, even if they have different coefficients.

For instance, \(18 y^{2}\) and \(6 y^{2}\) are like terms in our exercise because they are both terms with \(y\) raised to the second power. Similarly, \(12 x y\) and \(x y\) are like terms as they both have the same variables \(x\) and \(y\). To combine them, simply add or subtract their coefficients as instructed by their operations: \(18 y^{2} - 6 y^{2}\) becomes \(12 y^{2}\) and \(12 x y - x y\) becomes \(13 x y\).

Simplifying Algebraic Expressions

The purpose of simplifying algebraic expressions is to make them as uncomplicated as possible. After applying the distributive property and combining like terms, we are left with a more straightforward expression that is easier to work with. The initial complex expression \( (18 y^{2}-12 x y)-(6 y^{2}+x y-a) \) has been simplified to \(12 y^{2} - 13 x y + a\).

The simplification process involves several actions: removing parentheses, combining like terms, and ordering terms in a standardized way, often from the highest to the lowest power of any variables present. This process does not only make it easier to understand what the expression represents, but it also sets up a foundation for solving more complex equations later on.

Polynomial Operations

Polynomial operations, including subtraction as shown in our example, follow specific rules that apply to polynomials of any degree. A polynomial is an algebraic expression consisting of variables and coefficients, constructed with operations of addition, subtraction, multiplication, and natural number exponents of variables.

Subtracting polynomials, as in the exercise \( (18 y^{2}-12 x y)-(6 y^{2}+x y-a) \) involves performing subtraction on each corresponding term. It is essential to keep track of the sign in front of each term, as it affects the operation. A common mistake is to overlook changing the sign of the terms when distributing a negative over a polynomial, so practicing these operations carefully can help to avoid errors in more complex algebraic manipulations.