Find each product. (r-3s+t)(2r-s+t)

Verified step by step guidance

1. Identify the two binomials to be multiplied: \((r - 3s + t)\) and \((2r - s + t)\).
2. Use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. This means multiplying \(r\) by each term in \((2r - s + t)\), then \(-3s\) by each term in \((2r - s + t)\), and finally \(t\) by each term in \((2r - s + t)\).
3. Write out all the products explicitly: \(r \times 2r\), \(r \times (-s)\), \(r \times t\), \((-3s) \times 2r\), \((-3s) \times (-s)\), \((-3s) \times t\), \(t \times 2r\), \(t \times (-s)\), and \(t \times t\).
4. Simplify each product by performing multiplication and applying any necessary algebraic rules (like \(s \times s = s^2\) and \(t \times t = t^2\)).
5. Combine all the simplified terms into a single expression and then combine like terms to write the final expanded product.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial. This process requires applying the distributive property to combine like terms and simplify the expression into a single polynomial.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by a sum or difference inside parentheses, which is essential when expanding products of polynomials.

Combining Like Terms

After multiplying polynomials, you often get several terms with the same variables raised to the same powers. Combining like terms means adding or subtracting their coefficients to simplify the expression into its simplest form.