If \((\mathrm{t} - 8)\) is a factor of t^2 - kt - 48, then k =

1. Translate the problem requirements: If \(\mathrm{(t-8)}\) is a factor of the polynomial \(\mathrm{t^2 - kt - 48}\), this means when we substitute \(\mathrm{t = 8}\) into the polynomial, the result must equal zero
2. Apply the factor-root relationship: Since \(\mathrm{(t-8)}\) is a factor, \(\mathrm{t = 8}\) must be a root, so substitute \(\mathrm{t = 8}\) into the polynomial and set it equal to zero
3. Solve for the unknown coefficient: Use the equation from step 2 to find the value of k by performing basic algebraic operations
4. Verify the solution: Check that our value of k makes mathematical sense by confirming the factorization works

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is really asking us. We have a polynomial expression \(\mathrm{t^2 - kt - 48}\), and we're told that \(\mathrm{(t - 8)}\) is a factor of this polynomial.

What does it mean for \(\mathrm{(t - 8)}\) to be a factor? Think of it like this: if you have a number that divides evenly into another number, the first number is a factor of the second. Similarly, if \(\mathrm{(t - 8)}\) divides evenly into our polynomial \(\mathrm{t^2 - kt - 48}\), then \(\mathrm{(t - 8)}\) is a factor.

Here's the key insight: if \(\mathrm{(t - 8)}\) is a factor of the polynomial, then when we substitute \(\mathrm{t = 8}\) into the polynomial, the result must be zero. This is because when \(\mathrm{t = 8}\), the factor \(\mathrm{(t - 8)}\) becomes \(\mathrm{(8 - 8) = 0}\), and anything multiplied by 0 equals 0.

Process Skill: TRANSLATE - Converting the factor relationship into a mathematical condition

2. Apply the factor-root relationship

Now let's use what we discovered in step 1. Since \(\mathrm{(t - 8)}\) is a factor, we know that \(\mathrm{t = 8}\) must make our polynomial equal to zero.

Let's substitute \(\mathrm{t = 8}\) into the polynomial \(\mathrm{t^2 - kt - 48}\):

When \(\mathrm{t = 8}\):
\(\mathrm{(8)^2 - k(8) - 48 = 0}\)

Let's calculate this step by step:
• \(\mathrm{(8)^2 = 64}\)
• \(\mathrm{k(8) = 8k}\)
• So we have: \(\mathrm{64 - 8k - 48 = 0}\)

Simplifying the numbers: \(\mathrm{64 - 48 = 16}\)
So our equation becomes: \(\mathrm{16 - 8k = 0}\)

3. Solve for the unknown coefficient

Now we have a simple equation to solve: \(\mathrm{16 - 8k = 0}\)

Let's solve for k:
\(\mathrm{16 - 8k = 0}\)
\(\mathrm{16 = 8k}\) (adding 8k to both sides)
\(\mathrm{k = 16 ÷ 8}\) (dividing both sides by 8)
\(\mathrm{k = 2}\)

So \(\mathrm{k = 2}\).

4. Verify the solution

Let's check our answer by substituting \(\mathrm{k = 2}\) back into the original polynomial and confirming that \(\mathrm{(t - 8)}\) is indeed a factor.

With \(\mathrm{k = 2}\), our polynomial becomes: \(\mathrm{t^2 - 2t - 48}\)

If \(\mathrm{(t - 8)}\) is truly a factor, then we should be able to write:
\(\mathrm{t^2 - 2t - 48 = (t - 8) × (some\ other\ factor)}\)

Let's check by substituting \(\mathrm{t = 8}\):
\(\mathrm{(8)^2 - 2(8) - 48 = 64 - 16 - 48 = 0}\) ✓

This confirms our answer is correct.

Final Answer

The value of k is 2, which corresponds to answer choice (C).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "factor" means: Students might think that if \(\mathrm{(t - 8)}\) is a factor, they need to divide the polynomial by \(\mathrm{(t - 8)}\) using long division, rather than recognizing the simpler approach that if \(\mathrm{(t - 8)}\) is a factor, then \(\mathrm{t = 8}\) must be a root (make the polynomial equal zero).

2. Sign confusion with the factor: Students may incorrectly think that since we have \(\mathrm{(t - 8)}\) as a factor, we should substitute \(\mathrm{t = -8}\) instead of \(\mathrm{t = 8}\) into the polynomial, confusing the relationship between factors and roots.

3. Misinterpreting the problem setup: Students might try to factor the polynomial \(\mathrm{t^2 - kt - 48}\) first without realizing they need to find k first, leading them to attempt factoring with an unknown coefficient.

Errors while executing the approach

1. Arithmetic errors in substitution: When substituting \(\mathrm{t = 8}\), students commonly make mistakes like calculating \(\mathrm{8^2 = 16}\) instead of 64, or incorrectly computing \(\mathrm{64 - 48}\) as something other than 16.

2. Sign errors in the equation setup: Students may incorrectly write the substituted equation as \(\mathrm{64 + 8k - 48 = 0}\) instead of \(\mathrm{64 - 8k - 48 = 0}\), confusing the sign of the middle term.

3. Algebraic manipulation errors: When solving \(\mathrm{16 - 8k = 0}\), students might incorrectly isolate k by writing \(\mathrm{k = 16 + 8 = 24}\) or \(\mathrm{k = 8 - 16 = -8}\), rather than properly moving terms and dividing.

Errors while selecting the answer

1. Verification oversight: Students might arrive at \(\mathrm{k = 2}\) but select a different answer choice due to not double-checking their work or misreading the answer choices, especially confusing positive and negative values.

2. Calculation confirmation error: Even after finding \(\mathrm{k = 2}\), students might incorrectly verify by substituting back into the original polynomial and making arithmetic errors that lead them to think their answer is wrong, causing them to select a different choice.