If x³ + kx + 12 is completely divisible by (x + 3), then the value of k is:

Q: If x³ + kx + 12 is completely divisible by (x + 3), then the value of k is:

To determine the value of $ k $ such that the polynomial $ x^3 + kx + 12 $ is completely divisible by $ (x + 3) $, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial $ f(x) $ is divisible by $ (x - a) $, then $ f(a) = 0 $. In this case, the divisor is $ (x + 3) $, which can be rewritten as $ (x - (-3)) $. Therefore, we need to find the value of $ k $ such that: \[ f(-3) = (-3)^3 + k(-3) + 12 = 0 \] Calculating each term: 1. $ (-3)^3 = -27 $ 2. $ k(-3) = -3k $ 3. The constant term is $ 12 $ Substituting these into the equation: \[ -27 - 3k + 12 = 0 \] Simplifying the equation: \[ -15 - 3k = 0 \] Solving for $ k $: \[ -3k = 15 \] \[ k = -5 \] Thus, the value of $ k $ that makes the polynomial divisible by $ (x + 3) $ is $-5$. Therefore, Option 3 is correct. ### Explanation of Incorrect Options: 1. **Option 1: 5** - If $ k = 5 $, substituting into the equation gives: \[ -27 - 3(5) + 12 = -27 - 15 + 12 = -30 \neq 0 \] - Therefore, the polynomial is not divisible by $ (x + 3) $. 2. **Option 2: 4** - If $ k = 4 $, substituting into the equation gives: \[ -27 - 3(4) + 12 = -27 - 12 + 12 = -27 \neq 0 \] - Therefore, the polynomial is not divisible by $ (x + 3) $. 3. **Option 4: 3** - If $ k = 3 $, substituting into the equation gives: \[ -27 - 3(3) + 12 = -27 - 9 + 12 = -24 \neq 0 \] - Therefore, the polynomial is not divisible by $ (x + 3) $. In conclusion, only Option 3, where $ k = -5 $, satisfies the condition for divisibility by $ (x + 3) $.

Question

Accepted Answer

To determine the value of $ k $ such that the polynomial $ x^3 + kx + 12 $ is completely divisible by $ (x + 3) $, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial $ f(x) $ is divisible by $ (x - a) $, then $ f(a) = 0 $.

In this case, the divisor is $ (x + 3) $, which can be rewritten as $ (x - (-3)) $. Therefore, we need to find the value of $ k $ such that:

$$ f(-3) = (-3)^3 + k(-3) + 12 = 0 $$

Calculating each term:

1. $ (-3)^3 = -27 $
2. $ k(-3) = -3k $
3. The constant term is $ 12 $

Substituting these into the equation:

$$ -27 - 3k + 12 = 0 $$

Simplifying the equation:

$$ -15 - 3k = 0 $$

Solving for $ k $:

$$ -3k = 15 $$

$$ k = -5 $$

Thus, the value of $ k $ that makes the polynomial divisible by $ (x + 3) $ is $-5$. Therefore, Option 3 is correct.

### Explanation of Incorrect Options:

1. **Option 1: 5**
   - If $ k = 5 $, substituting into the equation gives:
     $$ -27 - 3(5) + 12 = -27 - 15 + 12 = -30 \neq 0 $$
   - Therefore, the polynomial is not divisible by $ (x + 3) $.

2. **Option 2: 4**
   - If $ k = 4 $, substituting into the equation gives:
     $$ -27 - 3(4) + 12 = -27 - 12 + 12 = -27 \neq 0 $$
   - Therefore, the polynomial is not divisible by $ (x + 3) $.

3. **Option 4: 3**
   - If $ k = 3 $, substituting into the equation gives:
     $$ -27 - 3(3) + 12 = -27 - 9 + 12 = -24 \neq 0 $$
   - Therefore, the polynomial is not divisible by $ (x + 3) $.

In conclusion, only Option 3, where $ k = -5 $, satisfies the condition for divisibility by $ (x + 3) $.