In Exercises 15–58, find each product. (7x^2−2)(3x^2−5)

Verified step by step guidance

1. Identify the expression to be multiplied: $(7x^2-2)(3x^2-5)$.
2. Apply the distributive property (also known as the FOIL method for binomials) to expand the expression: Multiply each term in the first binomial by each term in the second binomial.
3. First, multiply the first terms: $7x^2\times 3x^2$.
4. Next, multiply the outer terms: $7x^2\times -5$.
5. Then, multiply the inner terms: $-2\times 3x^2$, and finally, multiply the last terms: $-2\times -5$.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in the first polynomial to every term in the second polynomial. This process is often referred to as the FOIL method for binomials, but it generalizes to polynomials of any degree. The goal is to combine like terms after distribution to simplify the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This principle is fundamental in algebra and is used extensively in polynomial multiplication. It allows us to multiply a single term by a sum or difference, ensuring that each component is accounted for in the final expression.

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 crucial after multiplying polynomials, as it helps to condense the expression into its simplest form, making it easier to interpret and work with.