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The problem asks for the remainders when the function \( f(x) = 7x^2 - 6x + 3 \) is divided by different linear factors.

To find the remainder when dividing by a linear divisor of the form \( x - c \), we can use the **Remainder Theorem**. According to this theorem, if a polynomial \( f(x) \) is divided by \( x - c \), the remainder is \( f(c) \). This means that we can find the remainder by evaluating \( f(x) \) at the value of \( c \).

Let's proceed with the divisions:

1. **When \( f(x) \) is divided by \( x + 2 \):**

   This is the same as dividing by \( x - (-2) \), so we need to evaluate \( f(-2) \):
   \[
   f(-2) = 7(-2)^2 - 6(-2) + 3 = 7(4) + 12 + 3 = 28 + 12 + 3 = 43
   \]
   Remainder: **43**

2. **When \( f(x) \) is divided by \( x + 1 \):**

   This is the same as dividing by \( x - (-1) \), so we need to evaluate \( f(-1) \):
   \[
   f(-1) = 7(-1)^2 - 6(-1) + 3 = 7(1) + 6 + 3 = 7 + 6 + 3 = 16
   \]
   Remainder: **16**

3. **When \( f(x) \) is divided by \( x \):**

   This is the same as dividing by \( x - 0 \), so we need to evaluate \( f(0) \):
   \[
   f(0) = 7(0)^2 - 6(0) + 3 = 3
   \]
   Remainder: **3**

4. **When \( f(x) \) is divided by \( x - 1 \):**

   This is the same as dividing by \( x - 1 \), so we need to evaluate \( f(1) \):
   \[
   f(1) = 7(1)^2 - 6(1) + 3 = 7(1) - 6 + 3 = 7 - 6 + 3 = 4
   \]
   Remainder: **4**

5. **When \( f(x) \) is divided by \( x - 2 \):**

   This is the same as dividing by \( x - 2 \), so we need to evaluate \( f(2) \):
   \[
   f(2) = 7(2)^2 - 6(2) + 3 = 7(4) - 12 + 3 = 28 - 12 + 3 = 19
   \]
   Remainder: **19**

### Summary of Remainders:
- Divided by \( x + 2 \): remainder is **43**
- Divided by \( x + 1 \): remainder is **16**
- Divided by \( x \): remainder is **3**
- Divided by \( x - 1 \): remainder is **4**
- Divided by \( x - 2 \): remainder is **19**

Would you like further details or clarifications?

Here are five related questions to further explore:
1. How does the Remainder Theorem simplify division of polynomials?
2. Can we use synthetic division instead of the Remainder Theorem here?
3. What is the relationship between division and roots of a polynomial?
4. How would the remainders change if we modified the coefficients of \( f(x) \)?
5. What happens if we divide a higher degree polynomial by a linear factor?

**Tip:** The Remainder Theorem is a quick way to find remainders without performing long division—simply substitute the divisor's root into the polynomial!

Let f(x) = 7x^2 - 6x + 3. When f(x) is divided by x + 2, x + 1, x, x - 1, and x - 2, what are the remainders?

Find the remainders when dividing the quadratic function f(x) = 7x^2 - 6x + 3 by various linear factors using the Remainder Theorem. Suitable for high school algebra students.

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