In Exercises 83–90, perform the indicated operation or operations. (3x+5)(2x−9)−(7x−2)(x−1)

Verified step by step guidance

1. First, apply the distributive property to expand the expression \((3x+5)(2x−9)\). Multiply each term in the first binomial by each term in the second binomial.
2. Calculate \(3x \cdot 2x\), \(3x \cdot (-9)\), \(5 \cdot 2x\), and \(5 \cdot (-9)\) to expand the first part.
3. Next, apply the distributive property to expand the expression \((7x−2)(x−1)\). Multiply each term in the first binomial by each term in the second binomial.
4. Calculate \(7x \cdot x\), \(7x \cdot (-1)\), \(-2 \cdot x\), and \(-2 \cdot (-1)\) to expand the second part.
5. Combine like terms from both expanded expressions and simplify the expression by subtracting the second expanded expression from the first.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process, often referred to as the FOIL method for binomials, ensures that all combinations of terms are accounted for. For example, in the expression (3x + 5)(2x - 9), you would multiply 3x by 2x, 3x by -9, 5 by 2x, and 5 by -9, combining like terms afterward.

Combining Like Terms

Combining like terms is a fundamental algebraic process that simplifies expressions by merging terms that have the same variable raised to the same power. For instance, in the expression 6x^2 + 3x - 2x^2 + 4, the like terms 6x^2 and -2x^2 can be combined to yield 4x^2. This step is crucial for simplifying the result of polynomial operations.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing for the multiplication of a single term by a sum or difference. This property is essential when expanding expressions, such as when subtracting one polynomial from another. In the given problem, applying the distributive property helps to correctly handle the subtraction of the second polynomial from the first, ensuring accurate results.