Choose the correct simplification of [tex]$9 x^2\left(4 x+2 x^2-1\right)$[/tex].
A. [tex]$18 x^4+36 x^3-9 x^2$[/tex]
B. [tex]$18 x^4-36 x^3+9 x^2$[/tex]
C. [tex]$36 x^4+18 x^3-9 x^2$[/tex]
D. [tex]$36 x^4-13 x^3+9 x^2$[/tex]
Answer (1)
Distribute 9 x 2 to each term inside the parenthesis: 9 x 2 ∗ 4 x + 9 x 2 ∗ 2 x 2 − 9 x 2 ∗ 1 .
Simplify each term: 36 x 3 + 18 x 4 − 9 x 2 .
Rearrange the terms in decreasing order of exponents: 18 x 4 + 36 x 3 − 9 x 2 .
The correct simplification is 18 x 4 + 36 x 3 − 9 x 2 .
Explanation
Understanding the Problem We are given the expression 9 x 2 ( 4 x + 2 x 2 − 1 ) and four possible simplifications. We need to choose the correct simplification.
Distributing To simplify the expression, we need to distribute 9 x 2 to each term inside the parenthesis: 9 x 2 ( 4 x + 2 x 2 − 1 ) = 9 x 2 ⋅ 4 x + 9 x 2 ⋅ 2 x 2 + 9 x 2 ⋅ ( − 1 )
Simplifying Each Term Now, we simplify each term: 9 x 2 ⋅ 4 x = 36 x 3 9 x 2 ⋅ 2 x 2 = 18 x 4 9 x 2 ⋅ ( − 1 ) = − 9 x 2 So, the expression becomes: 36 x 3 + 18 x 4 − 9 x 2
Rearranging Terms and Conclusion Finally, we rearrange the terms in decreasing order of exponents: 18 x 4 + 36 x 3 − 9 x 2 Comparing this with the given options, we see that the correct simplification is 18 x 4 + 36 x 3 − 9 x 2 .
Examples
Understanding polynomial simplification is crucial in various fields, such as engineering and physics, where complex equations need to be solved efficiently. For example, when designing a bridge, engineers use polynomial equations to model the load distribution and structural integrity. Simplifying these equations allows for easier analysis and accurate predictions of the bridge's behavior under different conditions. Similarly, in physics, simplifying polynomial expressions helps in modeling projectile motion or calculating energy levels in quantum mechanics, making complex problems more manageable and understandable.
