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The distributive property is a fundamental concept in algebra helping with operations like polynomial subtraction or addition. When subtraction is involved, like in this exercise, the distributive property requires us to carefully apply a change to each term. In this case, distributing a negative sign across the terms involves flipping each sign in the second polynomial. This means if you have an expression like

• \(-6y^2 - 6y - 13\)

by distributing the negative sign, each term becomes positive, transforming it into

• \(6y^2 + 6y + 13\)
.

By changing the signs accurately, you ensure that subsequent steps like combining terms are correct. This step lays the groundwork by aligning the signs appropriately for when we subtract polynomials.

Once you've applied the distributive property and adjusted the signs, the next step is combining like terms. This involves grouping terms that have the same variables raised to the same power. For example, backward in our task, we have terms like

• \(3y^2\) and \(6y^2\)
• \(-4y\) and \(6y\)
• \(7\) and \(13\)

In this exercise, like terms such as \(3y^2\) and \(-6y^2\) are combined to form \(-3y^2\). Similarly, for linear terms, \(-4y\) and \(6y\) combine to result in \(2y\). Finally, all constant numbers, like \(7\) and \(13\), are combined to give \(-6\). This process simplifies the expression by merging terms into a more concise form, making the expression easier to interpret and manage.

An algebraic expression, like the ones in our task, consists of variables, numbers, and arithmetic operations. These expressions form the backbone of algebra, providing a way to describe quantities and relationships algebraically. The exercise demonstrates polynomial expressions, which specifically include terms with variables raised to whole number exponents, such as

• \(3y^2\) and \(-6y^2\)
• \(-4y\) and \(6y\)
• constant terms like \(7\) and \(-13\)

Understanding how to manipulate these expressions with operations like subtraction enables solving more complex algebraic problems. It showcases skills such as re-arranging and simplifying terms. Polynomials are just one type of algebraic expression, but learning how to handle these gives you a strong foundation for engaging with algebra in various mathematical fields.