Expand and simplify (x + 9)(x – 2)
The Correct Answer and Explanation is :
To expand and simplify the expression ((x + 9)(x – 2)), we will use the distributive property, commonly known as the FOIL method for binomials, which stands for First, Outer, Inner, and Last.
- First: Multiply the first terms of each binomial:
[
x \cdot x = x^2
] - Outer: Multiply the outer terms:
[
x \cdot (-2) = -2x
] - Inner: Multiply the inner terms:
[
9 \cdot x = 9x
] - Last: Multiply the last terms of each binomial:
[
9 \cdot (-2) = -18
]
Now, combine all these results:
[
x^2 – 2x + 9x – 18
]
Next, we combine the like terms (-2x) and (9x):
[
x^2 + 7x – 18
]
Thus, the expanded and simplified form of the expression ((x + 9)(x – 2)) is:
[
\boxed{x^2 + 7x – 18}
]
Explanation
Expanding binomials like ((x + 9)(x – 2)) involves distributing each term in the first binomial by each term in the second binomial. This method ensures that we account for all possible products that arise from combining the two expressions.
The process begins with the “First” terms, which give us the leading term of the quadratic, (x^2). Then, the “Outer” and “Inner” terms yield the linear components of the expression, contributing to the (x) term. Finally, the “Last” multiplication produces the constant term.
Once all products are calculated, combining like terms is essential to simplify the expression fully. In this case, (-2x) and (9x) are like terms and can be combined into (7x).
The final result, (x^2 + 7x – 18), is a quadratic expression that can be useful in various applications, such as graphing parabolas or solving equations. Understanding this expansion technique lays a foundation for working with polynomials and algebraic expressions more generally.