Polynomial division worksheet using the Remainder Theorem, where students solve problems and color by the remainder number.

Math worksheet titled "The Remainder Theorem Color by Number" with polynomial division problems and color instructions.

JPG 1700×2200 120.9 KB Free · Personal Use

Quality Assured by Worksheets Library Team

Reviewed for educational accuracy and age-appropriateness

ID: #878037

✨ Generate New Worksheet

⭐

Show Answer Key & Explanations Step-by-step solution for: The Remainder Theorem Color by Number

▼

To solve the problems using the Remainder Theorem, we need to evaluate each polynomial at the value of \( x \) that makes the divisor zero. The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (x - c) \), the remainder is \( f(c) \).

Let's solve each problem step by step:

---

Problem 1:

\[
(5x^3 + 25x^2 + x + 5) \div (x + 4)
\]
- Divisor: \( x + 4 \). Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Evaluate \( f(x) = 5x^3 + 25x^2 + x + 5 \) at \( x = -4 \):
\[
f(-4) = 5(-4)^3 + 25(-4)^2 + (-4) + 5
\]
\[
= 5(-64) + 25(16) - 4 + 5
\]
\[
= -320 + 400 - 4 + 5
\]
\[
= 81
\]
- Remainder: \( 81 \).

---

Problem 2:

\[
(x^4 + 16x^3 + 94x^2 + 240x + 225) \div (x + 4)
\]
- Divisor: \( x + 4 \). Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Evaluate \( f(x) = x^4 + 16x^3 + 94x^2 + 240x + 225 \) at \( x = -4 \):
\[
f(-4) = (-4)^4 + 16(-4)^3 + 94(-4)^2 + 240(-4) + 225
\]
\[
= 256 + 16(-64) + 94(16) - 960 + 225
\]
\[
= 256 - 1024 + 1504 - 960 + 225
\]
\[
= 0
\]
- Remainder: \( 0 \).

---

Problem 3:

\[
(x^4 + 4x^3 - 13x^2 - 28x + 60) \div (x + 4)
\]
- Divisor: \( x + 4 \). Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Evaluate \( f(x) = x^4 + 4x^3 - 13x^2 - 28x + 60 \) at \( x = -4 \):
\[
f(-4) = (-4)^4 + 4(-4)^3 - 13(-4)^2 - 28(-4) + 60
\]
\[
= 256 + 4(-64) - 13(16) + 112 + 60
\]
\[
= 256 - 256 - 208 + 112 + 60
\]
\[
= 52
\]
- Remainder: \( 52 \).

---

Problem 4:

\[
(x^4 + x^3 - 15x^2 - 3x + 67) \div (x + 3)
\]
- Divisor: \( x + 3 \). Set \( x + 3 = 0 \Rightarrow x = -3 \).
- Evaluate \( f(x) = x^4 + x^3 - 15x^2 - 3x + 67 \) at \( x = -3 \):
\[
f(-3) = (-3)^4 + (-3)^3 - 15(-3)^2 - 3(-3) + 67
\]
\[
= 81 - 27 - 15(9) + 9 + 67
\]
\[
= 81 - 27 - 135 + 9 + 67
\]
\[
= 15
\]
- Remainder: \( 15 \).

---

Problem 5:

\[
(x^4 + 2x^3 - 23x^2 - 24x + 144) \div (x - 2)
\]
- Divisor: \( x - 2 \). Set \( x - 2 = 0 \Rightarrow x = 2 \).
- Evaluate \( f(x) = x^4 + 2x^3 - 23x^2 - 24x + 144 \) at \( x = 2 \):
\[
f(2) = (2)^4 + 2(2)^3 - 23(2)^2 - 24(2) + 144
\]
\[
= 16 + 2(8) - 23(4) - 48 + 144
\]
\[
= 16 + 16 - 92 - 48 + 144
\]
\[
= 36
\]
- Remainder: \( 36 \).

---

Problem 6:

\[
(6x^3 + 20x^2 - 15x + 9) \div (x + 4)
\]
- Divisor: \( x + 4 \). Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Evaluate \( f(x) = 6x^3 + 20x^2 - 15x + 9 \) at \( x = -4 \):
\[
f(-4) = 6(-4)^3 + 20(-4)^2 - 15(-4) + 9
\]
\[
= 6(-64) + 20(16) + 60 + 9
\]
\[
= -384 + 320 + 60 + 9
\]
\[
= 5
\]
- Remainder: \( 5 \).

---

Problem 7:

\[
(x^4 + 4x^3 - 12x^2 - 32x + 64) \div (x + 5)
\]
- Divisor: \( x + 5 \). Set \( x + 5 = 0 \Rightarrow x = -5 \).
- Evaluate \( f(x) = x^4 + 4x^3 - 12x^2 - 32x + 64 \) at \( x = -5 \):
\[
f(-5) = (-5)^4 + 4(-5)^3 - 12(-5)^2 - 32(-5) + 64
\]
\[
= 625 + 4(-125) - 12(25) + 160 + 64
\]
\[
= 625 - 500 - 300 + 160 + 64
\]
\[
= 49
\]
- Remainder: \( 49 \).

---

Problem 8:

\[
(3x^3 + 6x^2 + x + 2) \div (x + 1)
\]
- Divisor: \( x + 1 \). Set \( x + 1 = 0 \Rightarrow x = -1 \).
- Evaluate \( f(x) = 3x^3 + 6x^2 + x + 2 \) at \( x = -1 \):
\[
f(-1) = 3(-1)^3 + 6(-1)^2 + (-1) + 2
\]
\[
= 3(-1) + 6(1) - 1 + 2
\]
\[
= -3 + 6 - 1 + 2
\]
\[
= 4
\]
- Remainder: \( 4 \).

---

Problem 9:

\[
(8x^5 + 32x^4 + 5x + 20) \div (x + 4)
\]
- Divisor: \( x + 4 \). Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Evaluate \( f(x) = 8x^5 + 32x^4 + 5x + 20 \) at \( x = -4 \):
\[
f(-4) = 8(-4)^5 + 32(-4)^4 + 5(-4) + 20
\]
\[
= 8(-1024) + 32(256) - 20 + 20
\]
\[
= -8192 + 8192 - 20 + 20
\]
\[
= 0
\]
- Remainder: \( 0 \).

---

Problem 10:

\[
(9x^3 - 18x^2 + 2x - 4) \div (x - 1)
\]
- Divisor: \( x - 1 \). Set \( x - 1 = 0 \Rightarrow x = 1 \).
- Evaluate \( f(x) = 9x^3 - 18x^2 + 2x - 4 \) at \( x = 1 \):
\[
f(1) = 9(1)^3 - 18(1)^2 + 2(1) - 4
\]
\[
= 9 - 18 + 2 - 4
\]
\[
= -11
\]
- Remainder: \( -11 \).

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad 81 \quad (\text{Blue}) \\
2. & \quad 0 \quad (\text{Teal}) \\
3. & \quad 52 \quad (\text{Purple}) \\
4. & \quad 15 \quad (\text{Pink}) \\
5. & \quad 36 \quad (\text{Black}) \\
6. & \quad 5 \quad (\text{Blue}) \\
7. & \quad 49 \quad (\text{Teal}) \\
8. & \quad 4 \quad (\text{Purple}) \\
9. & \quad 0 \quad (\text{Pink}) \\
10. & \quad -11 \quad (\text{Black})
\end{aligned}
}
\]

Parent Tip: Review the logic above to help your child master the concept of remainder theorem worksheet.

Print Download