Simplify the following: $(8 x-6)\left(-2 x^2+4 x-3\right)$
Answer (1)
Distribute 8 x across the second polynomial: − 16 x 3 + 32 x 2 − 24 x .
Distribute − 6 across the second polynomial: 12 x 2 − 24 x + 18 .
Combine the two resulting expressions: − 16 x 3 + 32 x 2 − 24 x + 12 x 2 − 24 x + 18 .
Simplify by combining like terms: − 16 x 3 + 44 x 2 − 48 x + 18 .
Explanation
Understanding the Problem We are given the expression ( 8 x − 6 ) ( − 2 x 2 + 4 x − 3 ) to simplify. Our goal is to expand this product and combine like terms to obtain a simplified polynomial.
Distributing the First Term First, we distribute 8 x over the terms of the second polynomial:
8 x ∗ ( − 2 x 2 + 4 x − 3 ) = 8 x ∗ − 2 x 2 + 8 x ∗ 4 x + 8 x ∗ − 3 = − 16 x 3 + 32 x 2 − 24 x .
Distributing the Second Term Next, we distribute − 6 over the terms of the second polynomial:
− 6 ∗ ( − 2 x 2 + 4 x − 3 ) = − 6 ∗ − 2 x 2 + − 6 ∗ 4 x + − 6 ∗ − 3 = 12 x 2 − 24 x + 18 .
Adding the Expressions Now, we add the two resulting expressions:
( − 16 x 3 + 32 x 2 − 24 x ) + ( 12 x 2 − 24 x + 18 ) = − 16 x 3 + 32 x 2 − 24 x + 12 x 2 − 24 x + 18 .
Combining Like Terms Finally, we combine like terms:
− 16 x 3 + ( 32 x 2 + 12 x 2 ) + ( − 24 x − 24 x ) + 18 = − 16 x 3 + 44 x 2 − 48 x + 18 .
So, the simplified expression is − 16 x 3 + 44 x 2 − 48 x + 18 .
Final Answer Therefore, the simplified form of the given expression is − 16 x 3 + 44 x 2 − 48 x + 18 .
Examples
Polynomial simplification is a fundamental concept in algebra and is used extensively in various fields. For instance, engineers use polynomial expressions to model physical systems, such as the trajectory of a projectile or the behavior of electrical circuits. Simplifying these expressions allows for easier analysis and prediction of system behavior. In economics, polynomial functions can represent cost and revenue curves, and simplifying them helps in optimizing business decisions.
