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.
That means if there’s a remainder, write it as a fraction over the divisor.
---
Problem 1:
(-2h³ - 18h² - 14h + 8) ÷ (h + 1)
We’ll use
synthetic division with root = -1 (since h + 1 = 0 → h = -1)
Coefficients of dividend:
-2 (h³), -18 (h²), -14 (h), +8 (constant)
Set up synthetic division:
```
-1 | -2 -18 -14 8
↓ 2 16 -2
---------------------
-2 -16 2 6
```
Steps:
- Bring down -2.
- Multiply -2 × (-1) = 2 → add to next term: -18 + 2 = -16
- Multiply -16 × (-1) = 16 → add to next: -14 + 16 = 2
- Multiply 2 × (-1) = -2 → add to last: 8 + (-2) = 6 ← this is the
remainder
So quotient is:
-2h² -16h + 2, remainder =
6
Answer:
-2h² -16h + 2 + 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:
(3p³ -13p² +18p +13) ÷ (p - 6)
Use synthetic division with root = 6 (since p - 6 = 0 → p = 6)
Coefficients: 3, -13, 18, 13
```
6 | 3 -13 18 13
↓ 18 30 288
-------------------------
3 5 48 301
```
Steps:
- Bring down 3.
- 3 × 6 = 18 → -13 + 18 = 5
- 5 × 6 = 30 → 18 + 30 = 48
- 48 × 6 = 288 → 13 + 288 = 301 ← remainder
Quotient:
3p² + 5p + 48, remainder =
301
Answer:
3p² + 5p + 48 + 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:
(-2c³ -19c² -19c -13) ÷ (c + 6)
Root = -6 (since c + 6 = 0 → c = -6)
Coefficients: -2, -19, -19, -13
Synthetic division:
```
-6 | -2 -19 -19 -13
↓ 12 42 -138
---------------------------
-2 -7 23 -151
```
Steps:
- Bring down -2.
- -2 × (-6) = 12 → -19 + 12 = -7
- -7 × (-6) = 42 → -19 + 42 = 23
- 23 × (-6) = -138 → -13 + (-138) = -151 ← remainder
Quotient:
-2c² -7c + 23, remainder =
-151
Answer:
-2c² -7c + 23 + (-151)/(c + 6)
or written as:
-2c² -7c + 23 - 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. $-2c^2 - 7c + 23 - \frac{151}{c + 6}$
Parent Tip: Review the logic above to help your child master the concept of polynomial long division worksheet.
