E/P) 17 Find the remainder when $$ x^{3}+6 x^{2}+5 x-12 $$ is divided by $$ i x-2 $$, ii $$ x+3 $$. b Hence, or otherwise, find all the solutions to the equation $$ x^{3}+6 x^{2}+5 x-12=0 $$

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### Given Polynomial:
$$ f(x) = x^{3} + 6x^{2} + 5x - 12 $$

### Part (a):

#### (i) Remainder when $$ f(x) $$ is divided by $$ ix - 2 $$:
To find the remainder when $$ f(x) $$ is divided by $$ ix - 2 $$, we can use the **Remainder Theorem**. The theorem states that the remainder of a polynomial $$ f(x) $$ divided by $$ x - a $$ is $$ f(a) $$.

First, solve for $$ x $$ in the divisor $$ ix - 2 = 0 $$:
$$
ix - 2 = 0 \implies x = \frac{2}{i}
$$
Recall that $$ \frac{1}{i} = -i $$ (since $$ i \times (-i) = 1 $$). Therefore:
$$
x = \frac{2}{i} = -2i
$$

Now, evaluate $$ f(x) $$ at $$ x = -2i $$:
$$
f(-2i) = (-2i)^3 + 6(-2i)^2 + 5(-2i) - 12
$$
Calculate each term:
$$
(-2i)^3 = -8i^3 = -8(-i) = 8i \
6(-2i)^2 = 6 \times 4i^2 = 6 \times 4(-1) = -24 \
5(-2i) = -10i \
$$
Combine the terms:
$$
f(-2i) = 8i - 24 - 10i - 12 = (-2i) - 36 = -36 - 2i
$$
**Remainder:** $$-36 - 2i$$

#### (ii) Remainder when $$ f(x) $$ is divided by $$ x + 3 $$:
Using the Remainder Theorem again, set $$ x + 3 = 0 $$, so $$ x = -3 $$.

Evaluate $$ f(-3) $$:
$$
f(-3) = (-3)^3 + 6(-3)^2 + 5(-3) - 12 = -27 + 54 - 15 - 12 = 0
$$
**Remainder:** $$0$$

### Part (b): Finding All Solutions to $$ f(x) = 0 $$
Since the remainder is zero when $$ f(x) $$ is divided by $$ x + 3 $$, $$ x = -3 $$ is a root of the polynomial. We can factor $$ f(x) $$ as follows:

$$
f(x) = (x + 3)(x^{2} + 3x - 4)
$$

Now, solve the quadratic equation $$ x^{2} + 3x - 4 = 0 $$:
$$
x = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm 5}{2}
$$
This gives two solutions:
$$
x = \frac{-3 + 5}{2} = 1 \quad \text{and} \quad x = \frac{-3 - 5}{2} = -4
$$

### **All Solutions:**
$$
x = -3, \quad x = 1, \quad x = -4
$$

### **Final Answer:**

**a.**
1. The remainder when divided by $$ ix - 2 $$ is $$ -36 - 2i $$.
2. The remainder when divided by $$ x + 3 $$ is $$ 0 $$.

**b.**
All solutions to the equation $$ x^{3} + 6x^{2} + 5x - 12 = 0 $$ are:
$$
x = -3,\quad x = 1,\quad \text{and}\quad x = -4.
$$
**a.** 1. The remainder when divided by $$ ix - 2 $$ is $$ -36 - 2i $$. 2. The remainder when divided by $$ x + 3 $$ is $$ 0 $$. **b.** All solutions to the equation $$ x^{3} + 6x^{2} + 5x - 12 = 0 $$ are: $$ x = -3,\quad x = 1,\quad \text{and}\quad x = -4. $$

E/P) 17 Find the remainder when is divided by
,
ii .
b Hence, or otherwise, find all the solutions to the equation

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E/P) 17 Find the remainder when is divided by , ii . b Hence, or otherwise, find all the solutions to the equation