Remainder Theorem Worksheets (printable, online, answers, examples) - Free Printable

You're asked to use the Remainder Theorem to find the remainders of several polynomial divisions.

---

✔ What is the Remainder Theorem?

> If a polynomial \( f(x) \) is divided by \( (x - c) \), then the remainder is \( f(c) \).

So, for each problem:
- Identify the divisor: it’s in the form \( (x - c) \)
- Plug \( x = c \) into the polynomial
- The result is the remainder

---

Let’s solve each one step by step.

---

## 🔹 Problem 1:
\((4x^3 + 8x^2 - 3x - 6) ÷ (x + 2)\)

Note: \( x + 2 = x - (-2) \), so \( c = -2 \)

Let \( f(x) = 4x^3 + 8x^2 - 3x - 6 \)

Find \( f(-2) \):

\[
f(-2) = 4(-2)^3 + 8(-2)^2 - 3(-2) - 6
= 4(-8) + 8(4) + 6 - 6
= -32 + 32 + 6 - 6
= 0
\]

✔ Remainder = 0

---

## 🔹 Problem 2:
\((x^3 - 4x^2 + x - 4) ÷ (x - 3)\)

Here, \( c = 3 \)

Let \( f(x) = x^3 - 4x^2 + x - 4 \)

Find \( f(3) \):

\[
f(3) = (3)^3 - 4(3)^2 + 3 - 4
= 27 - 4(9) + 3 - 4
= 27 - 36 + 3 - 4
= -10
\]

✔ Remainder = -10

---

## 🔹 Problem 3:
\((4x^3 - 20x^2 + 25x - 125) ÷ (x - 6)\)

Here, \( c = 6 \)

Let \( f(x) = 4x^3 - 20x^2 + 25x - 125 \)

Find \( f(6) \):

\[
f(6) = 4(216) - 20(36) + 25(6) - 125
= 864 - 720 + 150 - 125
= (864 - 720) = 144; (150 - 125) = 25 → 144 + 25 = 169
\]

✔ Remainder = 169

---

## 🔹 Problem 4:
\((x^3 - 3x^2 + 2x - 6) ÷ (x - 2)\)

Here, \( c = 2 \)

Let \( f(x) = x^3 - 3x^2 + 2x - 6 \)

Find \( f(2) \):

\[
f(2) = 8 - 3(4) + 4 - 6
= 8 - 12 + 4 - 6
= (8 - 12) = -4; (4 - 6) = -2 → -4 + (-2) = -6
\]

✔ Remainder = -6

---

## 🔹 Problem 5:
\((4x^3 - 16x^2 + x - 4) ÷ (x - 5)\)

Here, \( c = 5 \)

Let \( f(x) = 4x^3 - 16x^2 + x - 4 \)

Find \( f(5) \):

\[
f(5) = 4(125) - 16(25) + 5 - 4
= 500 - 400 + 5 - 4
= 100 + 1 = 101
\]

✔ Remainder = 101

---

## 📝 Final Answers:

1. \((4x^3 + 8x^2 - 3x - 6) ÷ (x + 2)\) → 0
2. \((x^3 - 4x^2 + x - 4) ÷ (x - 3)\) → -10
3. \((4x^3 - 20x^2 + 25x - 125) ÷ (x - 6)\) → 169
4. \((x^3 - 3x^2 + 2x - 6) ÷ (x - 2)\) → -6
5. \((4x^3 - 16x^2 + x - 4) ÷ (x - 5)\) → 101

---

✔ You’ve successfully used the Remainder Theorem — no long division needed!

Let me know if you’d like to see synthetic division or factoring as well!