Let f(x)=4x2−3x+5f(x)=4x2-3x+5.
When f(x)f(x) is divided by x+2x+2 the remainder is:
When f(x)f(x) is divided by x+1x+1 the remainder is:
When f(x)f(x) is divided by xx the remainder is:
When f(x)f(x) is divided by x−1x-1 the remainder is:
When f(x)f(x) is divided by x−2x-2 the remainder is:

To find the remainder when the polynomial \( f(x) = 4x^2 - 3x + 5 \) is divided by different binomials, we can use the **Remainder Theorem**. The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). 

Let's calculate the remainder for each case:

1. **When \( f(x) \) is divided by \( x + 2 \):**
   Using the Remainder Theorem, \( f(-2) \):
   \[
   f(-2) = 4(-2)^2 - 3(-2) + 5 = 4(4) + 6 + 5 = 16 + 6 + 5 = 27
   \]
   So, the remainder is \( \mathbf{27} \).

2. **When \( f(x) \) is divided by \( x + 1 \):**
   Using the Remainder Theorem, \( f(-1) \):
   \[
   f(-1) = 4(-1)^2 - 3(-1) + 5 = 4(1) + 3 + 5 = 4 + 3 + 5 = 12
   \]
   So, the remainder is \( \mathbf{12} \).

3. **When \( f(x) \) is divided by \( x \):**
   Using the Remainder Theorem, \( f(0) \):
   \[
   f(0) = 4(0)^2 - 3(0) + 5 = 5
   \]
   So, the remainder is \( \mathbf{5} \).

4. **When \( f(x) \) is divided by \( x - 1 \):**
   Using the Remainder Theorem, \( f(1) \):
   \[
   f(1) = 4(1)^2 - 3(1) + 5 = 4(1) - 3 + 5 = 4 - 3 + 5 = 6
   \]
   So, the remainder is \( \mathbf{6} \).

5. **When \( f(x) \) is divided by \( x - 2 \):**
   Using the Remainder Theorem, \( f(2) \):
   \[
   f(2) = 4(2)^2 - 3(2) + 5 = 4(4) - 6 + 5 = 16 - 6 + 5 = 15
   \]
   So, the remainder is \( \mathbf{15} \).

### Summary of Remainders:
1. \( x + 2 \): remainder is 27
2. \( x + 1 \): remainder is 12
3. \( x \): remainder is 5
4. \( x - 1 \): remainder is 6
5. \( x - 2 \): remainder is 15

Would you like more details on any of the steps or have any questions?

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Here are 5 related questions to explore further:
1. What is the Remainder Theorem, and how is it applied in polynomial division?
2. How would you factorize the polynomial \( f(x) = 4x^2 - 3x + 5 \)?
3. How do you perform synthetic division to verify the remainders?
4. Can you determine the quotient when dividing \( f(x) \) by \( x + 2 \)?
5. How would you find the remainder if you divided \( f(x) \) by a quadratic binomial?

**Tip:** The Remainder Theorem is a quick and efficient way to find remainders without needing to perform full polynomial division.

Let f(x) = 4x^2 - 3x + 5. When f(x) is divided by x + 2 the remainder is: When f(x) is divided by x + 1 the remainder is: When f(x) is divided by x the remainder is: When f(x) is divided by x - 1 the remainder is: When f(x) is divided by x - 2 the remainder is:

Learn how to find the remainders when the polynomial f(x) = 4x^2 - 3x + 5 is divided by various binomials using the Remainder Theorem. This problem covers key algebraic concepts and is suitable for students in Grades 9-12.

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