Math worksheet for students to practice applying the Remainder Theorem and Factor Theorem to find remainders and determine factors of polynomials.

Let's solve each problem step by step using the Remainder Theorem and Factor Theorem.

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🔷 Remainder Theorem:

When a polynomial $ P(x) $ is divided by $ x - c $, the remainder is $ P(c) $.

🔷 Factor Theorem:

If $ P(c) = 0 $, then $ x - c $ is a factor of $ P(x) $.
So if the remainder $ R = 0 $, then the divisor is a factor. Otherwise, it is not.

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## ✔ Problem 1:
Divide $ x^4 + 4x^3 - 11x - 5 $ by $ x + 3 $

We use $ P(-3) $ because $ x + 3 = x - (-3) $, so $ c = -3 $

Given:
$$
P(-3) = (-3)^4 + 4(-3)^3 - 11(-3) - 5
$$
$$
= 81 + 4(-27) + 33 - 5
$$
$$
= 81 - 108 + 33 - 5
$$
$$
= (81 + 33) - (108 + 5) = 114 - 113 = 1
$$

✔ So, $ P(-3) = R = \boxed{1} $

Since remainder ≠ 0 → Not a factor

👉 Answer:
- $ P(-3) = R = \boxed{1} $
- Is it a factor? → No

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## ✔ Problem 2:
Divide $ x^3 + 5x^2 - 2x - 24 $ by $ x - 2 $

Use $ P(2) $ because $ x - 2 $ → $ c = 2 $

$$
P(2) = (2)^3 + 5(2)^2 - 2(2) - 24
$$
$$
= 8 + 5(4) - 4 - 24
$$
$$
= 8 + 20 - 4 - 24
$$
$$
= 28 - 28 = 0
$$

✔ So, $ P(2) = R = \boxed{0} $

Since remainder = 0 → It is a factor

👉 Answer:
- $ P(2) = \boxed{0} $
- Is it a factor? → Yes

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## ✔ Problem 3:
Divide $ 4x^3 - 12x^2 - x + 3 $ by $ x - 3 $

Use $ P(3) $ because $ x - 3 $ → $ c = 3 $

$$
P(3) = 4(3)^3 - 12(3)^2 - (3) + 3
$$
$$
= 4(27) - 12(9) - 3 + 3
$$
$$
= 108 - 108 - 3 + 3
$$
$$
= 0 + 0 = 0
$$

✔ So, $ P(3) = R = \boxed{0} $

Since remainder = 0 → It is a factor

👉 Answer:
- $ P(3) = 4(27) - 12(9) - 3 + 3 = 108 - 108 - 3 + 3 = 0 $
- $ P(3) = R = \boxed{0} $
- Is it a factor? → Yes

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✔ Final Answers:

#### 1.
$ P(-3) = R = \boxed{1} $ → No, not a factor

#### 2.
$ P(2) = \boxed{0} $ → Yes, it is a factor

#### 3.
$ P(3) = \boxed{0} $ → Yes, it is a factor

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📝 Summary Table:

| Problem | Divisor | Remainder $ R $ | Is it a factor? |
|--------|--------|------------------|------------------|
| 1 | $ x+3 $ | 1 | No |
| 2 | $ x-2 $ | 0 | Yes |
| 3 | $ x-3 $ | 0 | Yes |

Let me know if you'd like this filled in on the worksheet format!