https://file.aiquickdraw.com/mathbot/file/1728610975021mrdwmg8c.jpg

Let's solve the first part of the problem from the image. We are asked to **state the remainder** when \(x + 2\) divides each polynomial.

To solve this, we will use **synthetic division** or **remainder theorem**. The remainder theorem tells us that if a polynomial \(f(x)\) is divided by \(x - a\), the remainder is \(f(a)\).

Here, since we are dividing by \(x + 2\), we set \(x + 2 = 0\), giving \(x = -2\). We will substitute \(x = -2\) into each polynomial.

### 4. a) \(x^2 + 7x + 9\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = (-2)^2 + 7(-2) + 9 = 4 - 14 + 9 = -1
\]

**Remainder = -1**

---

### 4. b) \(6x^3 + 19x^2 + 11x - 11\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = 6(-2)^3 + 19(-2)^2 + 11(-2) - 11
\]
\[
= 6(-8) + 19(4) + 11(-2) - 11 = -48 + 76 - 22 - 11 = -5
\]

**Remainder = -5**

---

### 4. c) \(x^4 - 5x^2 + 4\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = (-2)^4 - 5(-2)^2 + 4 = 16 - 5(4) + 4 = 16 - 20 + 4 = 0
\]

**Remainder = 0**

---

### 4. d) \(x^4 - 2x^3 - 11x^2 + 10x - 2\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = (-2)^4 - 2(-2)^3 - 11(-2)^2 + 10(-2) - 2
\]
\[
= 16 - 2(-8) - 11(4) + 10(-2) - 2 = 16 + 16 - 44 - 20 - 2 = -34
\]

**Remainder = -34**

---

### 4. e) \(x^3 + 3x^2 - 10x + 6\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = (-2)^3 + 3(-2)^2 - 10(-2) + 6 = -8 + 3(4) + 20 + 6 = -8 + 12 + 20 + 6 = 30
\]

**Remainder = 30**

---

### 4. f) \(4x^4 + 12x^3 - 13x^2 - 33x + 18\)

Substitute \(x = -2\) into the polynomial:

\[
f(-2) = 4(-2)^4 + 12(-2)^3 - 13(-2)^2 - 33(-2) + 18
\]
\[
= 4(16) + 12(-8) - 13(4) - 33(-2) + 18 = 64 - 96 - 52 + 66 + 18 = 0
\]

**Remainder = 0**

---

Now, for question **5**, we will determine whether \(2x - 5\) is a factor of the given polynomials. Let me know if you want me to continue solving it or if you have any specific questions.

### Do you have any questions or want more details about the process?

Here are 5 related questions you might find interesting:
1. How does the remainder theorem help in polynomial division?
2. What is the synthetic division method, and how is it different from long division?
3. Can the remainder theorem be applied to non-linear divisors like \(x^2 + 2x\)?
4. Why does a remainder of 0 imply that a divisor is a factor of the polynomial?
5. How can we use the factor theorem to factor polynomials completely?

#### Tip:
Always remember that if the remainder is 0 when dividing by \(x - a\), it means \(x - a\) is a factor of the polynomial.

State the remainder when x + 2 is divided into each polynomial.

Learn how to find the remainder when dividing polynomials by x + 2 using the remainder theorem. This problem involves synthetic division and substitution methods, suitable for Grades 10-12 students studying algebra and polynomial functions.

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