Three example problems for dividing cubic polynomials by linear binomials.

Let’s solve each polynomial division step by step. We’ll use polynomial long division or synthetic division — but since the divisor is a binomial like (h + 1), we can also use substitution to check remainders, and then write the answer as quotient + remainder/divisor.

We are told: “Put remainders in fractional forms.” So if there’s a remainder, we write it as a fraction over the divisor.

---

Problem 1:

Divide $(-2h^3 - 18h^2 - 14h + 8) \div (h + 1)$

We can use synthetic division with root = -1 (since h + 1 = 0 → h = -1)

Set up coefficients:
-2 -18 -14 8
Bring down -2.

Multiply by -1: (-2)(-1) = 2 → add to next term: -18 + 2 = -16
Multiply by -1: (-16)(-1) = 16 → add to next: -14 + 16 = 2
Multiply by -1: (2)(-1) = -2 → add to last: 8 + (-2) = 6 ← REMAINDER

So quotient is: -2h² -16h + 2, remainder 6.

Answer:
$-2h^2 - 16h + 2 + \frac{6}{h+1}$

✔ Check: Let’s plug h = -1 into original polynomial to verify remainder:

Original: -2(-1)^3 -18(-1)^2 -14(-1) + 8
= -2(-1) -18(1) +14 +8
= 2 -18 +14 +8 = (2+14+8) -18 = 24 - 18 = 6 ✔️

---

Problem 2:

$(3p^3 - 13p^2 + 18p + 13) \div (p - 6)$

Use synthetic division with root = 6.

Coefficients: 3 -13 18 13

Bring down 3.

Multiply by 6: 3×6 = 18 → add to -13: 5
Multiply by 6: 5×6 = 30 → add to 18: 48
Multiply by 6: 48×6 = 288 → add to 13: 301 ← REMAINDER

Quotient: 3p² + 5p + 48, remainder 301.

Answer:
$3p^2 + 5p + 48 + \frac{301}{p-6}$

✔ Check: Plug p = 6 into original:

3*(216) -13*(36) +18*6 +13
= 648 - 468 + 108 + 13
= (648 + 108 + 13) - 468 = 769 - 468 = 301 ✔️

---

Problem 3:

$(-2c^3 - 19c^2 - 19c - 13) \div (c + 6)$

Root = -6 (since c + 6 = 0 → c = -6)

Coefficients: -2 -19 -19 -13

Bring down -2.

Multiply by -6: (-2)(-6) = 12 → add to -19: -7
Multiply by -6: (-7)(-6) = 42 → add to -19: 23
Multiply by -6: (23)(-6) = -138 → add to -13: -151 ← REMAINDER

Quotient: -2c² -7c + 23, remainder -151.

Answer:
$-2c^2 - 7c + 23 + \frac{-151}{c+6}$
or written as:
$-2c^2 - 7c + 23 - \frac{151}{c+6}$

✔ Check: Plug c = -6 into original:

-2*(-216) -19*(36) -19*(-6) -13
= 432 - 684 + 114 -13
= (432 + 114) - (684 + 13) = 546 - 697 = -151 ✔️

---

Final Answer:

1. $-2h^2 - 16h + 2 + \frac{6}{h+1}$
2. $3p^2 + 5p + 48 + \frac{301}{p-6}$
3. $-2c^2 - 7c + 23 - \frac{151}{c+6}$