Answer :
Sure! Let's perform synthetic division step-by-step for each given exercise.
### Exercise 27:
Divide [tex]\(4x^2 + 15x + 1\)[/tex] by [tex]\(x + 6\)[/tex].
Synthetic Division Steps:
1. Write down the coefficients of the dividend: [tex]\(4, 15, 1\)[/tex].
2. The divisor is [tex]\(x + 6\)[/tex], so we use [tex]\(-6\)[/tex] (the opposite of [tex]\(6\)[/tex]).
3. Perform synthetic division:
- Bring down the first coefficient: [tex]\(4\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(4\)[/tex], and add to the next coefficient: [tex]\(15 - 24 = -9\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(-9\)[/tex], and add to the next coefficient: [tex]\(1 + 54 = 55\)[/tex].
The quotient is [tex]\(4x - 9\)[/tex] and the remainder is [tex]\(55\)[/tex].
### Exercise 29:
Divide [tex]\(5x^2 - 17x - 12\)[/tex] by [tex]\(x - 4\)[/tex].
Synthetic Division Steps:
1. Write down the coefficients: [tex]\(5, -17, -12\)[/tex].
2. Use [tex]\(4\)[/tex] from the divisor [tex]\(x - 4\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(5\)[/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(5\)[/tex] and add: [tex]\(-17 + 20 = 3\)[/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(3\)[/tex] and add: [tex]\(-12 + 12 = 0\)[/tex].
The quotient is [tex]\(5x + 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
### Exercise 31:
Divide [tex]\(4 - 8x - 3x^2 - 5x^4\)[/tex] by [tex]\(x + 2\)[/tex].
Synthetic Division Steps:
1. The polynomial is [tex]\( -5x^4 - 3x^2 - 8x + 4\)[/tex]. Coefficients: [tex]\(-5, 0, -3, -8, 4\)[/tex].
2. Use [tex]\(-2\)[/tex] from [tex]\(x + 2\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(-5\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(-5\)[/tex], add: [tex]\(0 + 10 = 10\)[/tex].
- [tex]\(-2 \times 10 = -20\)[/tex]; add: [tex]\(-3 - 20 = -23\)[/tex].
- [tex]\(-2 \times -23 = 46\)[/tex]; add: [tex]\(-8 + 46 = 38\)[/tex].
- [tex]\(-2 \times 38 = -76\)[/tex]; add: [tex]\(4 - 76 = -72\)[/tex].
The quotient is [tex]\(-5x^3 + 10x^2 - 23x + 38\)[/tex], remainder is [tex]\(-72\)[/tex].
### Exercise 33:
Divide [tex]\(4x^5 - 25x^4 - 58x^3 + 232x^2 + 198x - 63\)[/tex] by [tex]\(x - 3\)[/tex].
Synthetic Division Result:
The quotient is [tex]\(4x^4 - 13x^3 - 97x^2 - 59x + 21\)[/tex] and the remainder is [tex]\(0\)[/tex].
### Exercise 35:
Divide [tex]\(x^5 + 32\)[/tex] by [tex]\(x + 2\)[/tex].
Synthetic Division Steps:
1. Coefficients: [tex]\(1, 0, 0, 0, 0, 32\)[/tex].
2. Use [tex]\(-2\)[/tex] from the divisor [tex]\(x + 2\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(1\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(0 - 2 = -2\)[/tex].
- [tex]\(-2 \times -2 = 4\)[/tex]; add: [tex]\(0 + 4 = 4\)[/tex].
- [tex]\(-2 \times 4 = -8\)[/tex]; add: [tex]\(0 - 8 = -8\)[/tex].
- [tex]\(-2 \times -8 = 16\)[/tex]; add: [tex]\(0 + 16 = 16\)[/tex].
- [tex]\(-2 \times 16 = -32\)[/tex]; add: [tex]\(32 - 32 = 0\)[/tex].
The quotient is [tex]\(x^4 - 2x^3 + 4x^2 - 8x + 16\)[/tex] and the remainder is [tex]\(0\)[/tex].
I hope this helps! If you have any more questions, feel free to ask.
### Exercise 27:
Divide [tex]\(4x^2 + 15x + 1\)[/tex] by [tex]\(x + 6\)[/tex].
Synthetic Division Steps:
1. Write down the coefficients of the dividend: [tex]\(4, 15, 1\)[/tex].
2. The divisor is [tex]\(x + 6\)[/tex], so we use [tex]\(-6\)[/tex] (the opposite of [tex]\(6\)[/tex]).
3. Perform synthetic division:
- Bring down the first coefficient: [tex]\(4\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(4\)[/tex], and add to the next coefficient: [tex]\(15 - 24 = -9\)[/tex].
- Multiply [tex]\(-6\)[/tex] by [tex]\(-9\)[/tex], and add to the next coefficient: [tex]\(1 + 54 = 55\)[/tex].
The quotient is [tex]\(4x - 9\)[/tex] and the remainder is [tex]\(55\)[/tex].
### Exercise 29:
Divide [tex]\(5x^2 - 17x - 12\)[/tex] by [tex]\(x - 4\)[/tex].
Synthetic Division Steps:
1. Write down the coefficients: [tex]\(5, -17, -12\)[/tex].
2. Use [tex]\(4\)[/tex] from the divisor [tex]\(x - 4\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(5\)[/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(5\)[/tex] and add: [tex]\(-17 + 20 = 3\)[/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(3\)[/tex] and add: [tex]\(-12 + 12 = 0\)[/tex].
The quotient is [tex]\(5x + 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
### Exercise 31:
Divide [tex]\(4 - 8x - 3x^2 - 5x^4\)[/tex] by [tex]\(x + 2\)[/tex].
Synthetic Division Steps:
1. The polynomial is [tex]\( -5x^4 - 3x^2 - 8x + 4\)[/tex]. Coefficients: [tex]\(-5, 0, -3, -8, 4\)[/tex].
2. Use [tex]\(-2\)[/tex] from [tex]\(x + 2\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(-5\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(-5\)[/tex], add: [tex]\(0 + 10 = 10\)[/tex].
- [tex]\(-2 \times 10 = -20\)[/tex]; add: [tex]\(-3 - 20 = -23\)[/tex].
- [tex]\(-2 \times -23 = 46\)[/tex]; add: [tex]\(-8 + 46 = 38\)[/tex].
- [tex]\(-2 \times 38 = -76\)[/tex]; add: [tex]\(4 - 76 = -72\)[/tex].
The quotient is [tex]\(-5x^3 + 10x^2 - 23x + 38\)[/tex], remainder is [tex]\(-72\)[/tex].
### Exercise 33:
Divide [tex]\(4x^5 - 25x^4 - 58x^3 + 232x^2 + 198x - 63\)[/tex] by [tex]\(x - 3\)[/tex].
Synthetic Division Result:
The quotient is [tex]\(4x^4 - 13x^3 - 97x^2 - 59x + 21\)[/tex] and the remainder is [tex]\(0\)[/tex].
### Exercise 35:
Divide [tex]\(x^5 + 32\)[/tex] by [tex]\(x + 2\)[/tex].
Synthetic Division Steps:
1. Coefficients: [tex]\(1, 0, 0, 0, 0, 32\)[/tex].
2. Use [tex]\(-2\)[/tex] from the divisor [tex]\(x + 2\)[/tex].
3. Perform synthetic division:
- Bring down [tex]\(1\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(0 - 2 = -2\)[/tex].
- [tex]\(-2 \times -2 = 4\)[/tex]; add: [tex]\(0 + 4 = 4\)[/tex].
- [tex]\(-2 \times 4 = -8\)[/tex]; add: [tex]\(0 - 8 = -8\)[/tex].
- [tex]\(-2 \times -8 = 16\)[/tex]; add: [tex]\(0 + 16 = 16\)[/tex].
- [tex]\(-2 \times 16 = -32\)[/tex]; add: [tex]\(32 - 32 = 0\)[/tex].
The quotient is [tex]\(x^4 - 2x^3 + 4x^2 - 8x + 16\)[/tex] and the remainder is [tex]\(0\)[/tex].
I hope this helps! If you have any more questions, feel free to ask.
