Select the correct answer.
Kalid simplified a polynomial expression as shown.
$\begin{array}{ll}
& \left(6 x^3+8 x^2-7 x\right)-\left(2 x^2+3\right)(x-8) \\
\text { step 1 } & \left(6 x^3+8 x^2-7 x\right)-\left(2 x^3-16 x^2+3 x-24\right) \\
\text { step 2 } & 6 x^3+8 x^2-7 x-2 x^3-16 x^2+3 x-24 \\
\text { step 3 } & 4 x^3-8 x^2-4 x-24
\end{array}$
In which step did Kalid make a mistake?
A. step 1
B. step 2
C. step 3
D. Kalid did not make a mistake.
Answer (1)
Expand the product ( 2 x 2 + 3 ) ( x − 8 ) to get 2 x 3 − 16 x 2 + 3 x − 24 .
Distribute the negative sign to get 6 x 3 + 8 x 2 − 7 x − 2 x 3 + 16 x 2 − 3 x + 24 .
Combine like terms to get 4 x 3 + 24 x 2 − 10 x + 24 .
Kalid made a mistake in Step 3, so the answer is C. step 3 .
Explanation
Analyzing the Original Expression Let's analyze Kalid's simplification step by step to identify any mistakes.
Original Expression: ( 6 x 3 + 8 x 2 − 7 x ) − ( 2 x 2 + 3 ) ( x − 8 )
Checking Step 1 Step 1: Kalid expands the product ( 2 x 2 + 3 ) ( x − 8 ) :
( 2 x 2 + 3 ) ( x − 8 ) = 2 x 2 ( x − 8 ) + 3 ( x − 8 ) = 2 x 3 − 16 x 2 + 3 x − 24 So the expression becomes: ( 6 x 3 + 8 x 2 − 7 x ) − ( 2 x 3 − 16 x 2 + 3 x − 24 ) This matches Kalid's Step 1, so there is no mistake in Step 1.
Checking Step 2 Step 2: Kalid distributes the negative sign: ( 6 x 3 + 8 x 2 − 7 x ) − ( 2 x 3 − 16 x 2 + 3 x − 24 ) = 6 x 3 + 8 x 2 − 7 x − 2 x 3 + 16 x 2 − 3 x + 24 This matches Kalid's Step 2, so there is no mistake in Step 2.
Checking Step 3 Step 3: Kalid combines like terms: 6 x 3 − 2 x 3 + 8 x 2 + 16 x 2 − 7 x − 3 x + 24 = 4 x 3 + 24 x 2 − 10 x + 24 However, Kalid's Step 3 is: 4 x 3 − 8 x 2 − 4 x − 24 There is a mistake in combining the like terms. The correct combination should be 4 x 3 + 24 x 2 − 10 x + 24 , not 4 x 3 − 8 x 2 − 4 x − 24 .
Conclusion Therefore, Kalid made a mistake in Step 3.
Examples
Polynomial simplification is a fundamental skill in algebra and is used in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomial equations to model the load and stress distribution. Simplifying these equations allows them to analyze the structural integrity of the bridge more efficiently. Similarly, in computer graphics, polynomial functions are used to create curves and surfaces, and simplifying these functions can optimize rendering performance.
