In Exercises 15–58, find each product. (x+7)(x+3)

Verified step by step guidance

1. Identify the expression to be multiplied: $(x+7)(x+3)$.
2. Apply the distributive property (also known as the FOIL method for binomials): First, Outer, Inner, Last.
3. Multiply the First terms: $x\cdot x=x^2$.
4. Multiply the Outer terms: $x\cdot 3=3x$.
5. Multiply the Inner terms: $7\cdot x=7x$.
6. Multiply the Last terms: $7\cdot 3=21$.
7. Combine all the products: $x^2+3x+7x+21$.
8. Combine like terms: $3x+7x$ to simplify the expression.

Key Concepts

Distributive Property

The Distributive Property states that a(b + c) = ab + ac. This property allows us to multiply a single term by two or more terms inside parentheses. In the context of the given expression (x + 7)(x + 3), we will distribute each term in the first parentheses by each term in the second parentheses to find the product.

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 (x + 7)(x + 3), we will have several terms that can be combined, such as x², 10x, and constants, to arrive at a simplified polynomial.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to produce a new polynomial. In this case, we are multiplying two binomials, (x + 7) and (x + 3). The result will be a quadratic polynomial, which is a polynomial of degree 2, typically expressed in the standard form ax² + bx + c.