Find each product. See Examples 3–5. (r-3s+t)(2r-s+t)

Verified step by step guidance

1. Identify the expression to be expanded: \((r-3s+t)(2r-s+t)\).
2. Apply the distributive property (also known as the FOIL method for binomials) to expand the expression.
3. Multiply each term in the first parenthesis by each term in the second parenthesis: \(r(2r) + r(-s) + r(t) - 3s(2r) - 3s(-s) - 3s(t) + t(2r) + t(-s) + t(t)\).
4. Simplify each of the products: \(2r^2 - rs + rt - 6rs + 3s^2 - 3st + 2rt - st + t^2\).
5. Combine like terms to simplify the expression further.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process is often referred to as the FOIL method for binomials, which stands for First, Outside, Inside, Last. Understanding how to combine like terms after distribution is crucial for simplifying the resulting expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by a sum. This property is fundamental in algebra as it simplifies the process of expanding expressions. In the context of polynomials, it helps in systematically multiplying each term of one polynomial by each term of another.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is essential after multiplying polynomials, as it helps to condense the expression into its simplest form. Recognizing like terms is key to achieving a clear and concise final answer.