Find each product. See Example 5.(4r - 1) (7r + 2)

Verified step by step guidance

1. Identify the expression to be expanded: \((4r - 1)(7r + 2)\).
2. Apply the distributive property (also known as the FOIL method for binomials): First, Outer, Inner, Last.
3. Calculate the 'First' terms: \(4r \times 7r = 28r^2\).
4. Calculate the 'Outer' terms: \(4r \times 2 = 8r\).
5. Calculate the 'Inner' terms: \(-1 \times 7r = -7r\) and the 'Last' terms: \(-1 \times 2 = -2\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that states a(b + c) = ab + ac. It allows us to multiply a single term by two or more terms inside parentheses. In the context of the given expression, (4r - 1)(7r + 2), we will apply this property to distribute each term in the first parentheses across each term in the second parentheses.

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. After applying the distributive property to the expression (4r - 1)(7r + 2), we will likely end up with several terms that can be combined to form a simpler expression, making it easier to interpret the result.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to produce a new polynomial. In this case, we are multiplying a binomial (4r - 1) by another binomial (7r + 2). The result will be a polynomial of degree equal to the sum of the degrees of the two polynomials being multiplied, which is essential for understanding the structure of the resulting expression.