In Exercises 15–58, find each product. (7x²−2)(3x²−5)

Verified step by step guidance

1. Step 1: Recognize that the problem involves multiplying two binomials, (7x^2 - 2) and (3x^2 - 5). This can be solved using the distributive property, also known as the FOIL method (First, Outer, Inner, Last).
2. Step 2: Multiply the 'First' terms of each binomial: (7x^2) * (3x^2). This results in 21x^4.
3. Step 3: Multiply the 'Outer' terms of each binomial: (7x^2) * (-5). This results in -35x^2.
4. Step 4: Multiply the 'Inner' terms of each binomial: (-2) * (3x^2). This results in -6x^2.
5. Step 5: Multiply the 'Last' terms of each binomial: (-2) * (-5). This results in +10. Combine all these terms to form the expanded expression: 21x^4 - 35x^2 - 6x^2 + 10. Then simplify by combining like terms.

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 any polynomials. 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 property is fundamental in algebra as it allows for the multiplication of a single term by a sum or difference. In the context of polynomials, it helps in expanding expressions by ensuring that each term is multiplied correctly.

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 reduce the expression to its simplest form, making it easier to interpret and work with.