The product of a binomial and a trinomial is $x^3+3 x^2-x+2 x^2+6 x-2$. Which expression is equivalent to this product after it has been fully simplified?
A. $x^3+5 x^2+5 x-2$
B. $x^3+2 x^2+8 x-2$
C. $x^3+11 x^2-2$
D. $x^3+10 x^2-2$
Answer (1)
Identify like terms in the expression.
Combine the coefficients of the x 2 terms: 3 x 2 + 2 x 2 = 5 x 2 .
Combine the coefficients of the x terms: − x + 6 x = 5 x .
Write the simplified expression: x 3 + 5 x 2 + 5 x − 2 .
Explanation
Understanding the Problem We are given the expression x 3 + 3 x 2 − x + 2 x 2 + 6 x − 2 which is the product of a binomial and a trinomial. Our goal is to simplify this expression by combining like terms.
Identifying Like Terms First, let's identify the like terms. We have two x 2 terms: 3 x 2 and 2 x 2 . We also have two x terms: − x and 6 x . The other terms, x 3 and − 2 , are unique.
Combining Like Terms Now, let's combine the like terms. We add the coefficients of the x 2 terms: 3 + 2 = 5 . So, 3 x 2 + 2 x 2 = 5 x 2 . Next, we add the coefficients of the x terms: − 1 + 6 = 5 . So, − x + 6 x = 5 x .
Writing the Simplified Expression Finally, we write the simplified expression by combining all the terms: x 3 + 5 x 2 + 5 x − 2 .
Examples
Understanding how to simplify polynomial expressions is crucial in various fields, such as engineering and physics. For instance, when designing a bridge, engineers use polynomial equations to model the load distribution. Simplifying these equations allows them to efficiently calculate the stress and strain on different parts of the structure, ensuring its stability and safety. Similarly, in physics, simplifying polynomial expressions helps in modeling the trajectory of projectiles or the behavior of electrical circuits.
