Answered

Select the correct answer.

Kalid simplified a polynomial expression as shown.

[tex]\[
\begin{array}{ll}
& \left(6x^3 + 8x^2 - 7x\right) - \left(2x^2 + 3\right)(x - 8) \\
\text{Step 1} & \left(6x^3 + 8x^2 - 7x\right) - \left(2x^3 - 16x^2 + 3x - 24\right) \\
\text{Step 2} & 6x^3 + 8x^2 - 7x - 2x^3 - 16x^2 + 3x - 24 \\
\text{Step 3} & 4x^3 - 8x^2 - 4x - 24
\end{array}
\][/tex]

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 :

Let's analyze each step in detail to identify where the mistake might have occurred in simplifying the polynomial expression [tex]\((6x^3 + 8x^2 - 7x) - (2x^2 + 3)(x - 8)\)[/tex].

### Step 1

First, expand [tex]\((2x^2 + 3)(x - 8)\)[/tex]:

[tex]\[ (2x^2 + 3)(x - 8) = 2x^2 \cdot x + 2x^2 \cdot (-8) + 3 \cdot x + 3 \cdot (-8) \][/tex]

[tex]\[ = 2x^3 - 16x^2 + 3x - 24 \][/tex]

Now the expression becomes:

[tex]\[ (6x^3 + 8x^2 - 7x) - (2x^3 - 16x^2 + 3x - 24) \][/tex]

Kalid correctly expanded the expression and then wrote:

[tex]\[ \left(6 x^3 + 8 x^2 - 7 x\right) - \left(2 x^3 - 16 x^2 + 3 x - 24\right) \][/tex]

Therefore, Kalid's Step 1 is correct.

### Step 2

Distribute the negative sign and combine like terms:

[tex]\[ 6x^3 + 8x^2 - 7x - 2x^3 + 16x^2 - 3x + 24 \][/tex]

Combine like terms:

[tex]\[ (6x^3 - 2x^3) + (8x^2 + 16x^2) + (-7x - 3x) + 24 \][/tex]

[tex]\[ = 4x^3 + 24x^2 - 10x + 24 \][/tex]

After examining Kalid's Step 2:

[tex]\[ 6 x^3 + 8 x^2 - 7 x - 2 x^3 + 16 x^2 - 3 x - 24 \][/tex]

It appears he did not combine the constants correctly, as the last term should be [tex]\(+24\)[/tex] (positive) instead of [tex]\(-24\)[/tex] (negative). Therefore, Kalid made a mistake here.

### Step 3

Kalid's Step 3 gives:

[tex]\[ 4 x^3 - 8 x^2 - 4 x - 24 \][/tex]

But from our corrected combination in Step 2, we get:

[tex]\[ 4x^3 + 24x^2 - 10x + 24 \][/tex]

Therefore, something is definitely off given that the constants diverged in two different ways.

To maximize surprise, I can conclude that Kalid combined the constants incorrectly between Step 2 and 3. Factoring out the full expression with correct terms will still highlight the mistake in Step 1 cleanup for distribution.

### Conclusion

Kalid made a mistake in Step 2 when combining the constants. Thus, the correct answer is:

B. Step 2.