Answer :
Sure! Let's go through each of these polynomial long division problems step-by-step.
### Problem 19: Divide [tex]\(3x^4 - 4x^3 + 4x - 6\)[/tex] by [tex]\(3x + 5\)[/tex]
Step 1: Divide the first term
Take the first term of the dividend, [tex]\(3x^4\)[/tex], and divide it by the first term of the divisor, [tex]\(3x\)[/tex]. The result is [tex]\(x^3\)[/tex].
Step 2: Multiply and subtract
Multiply the entire divisor [tex]\(3x + 5\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(3x^4 + 5x^3\)[/tex].
Subtract this from the original dividend:
[tex]\((3x^4 - 4x^3) - (3x^4 + 5x^3) = -9x^3\)[/tex].
Step 3: Bring down the next term
Bring down the next term from the dividend, which is [tex]\(4x\)[/tex], to make [tex]\(-9x^3 + 4x\)[/tex].
Step 4: Repeat the process
- Divide [tex]\(-9x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-3x^2\)[/tex].
- Multiply the divisor by [tex]\(-3x^2\)[/tex] giving [tex]\(-9x^3 - 15x^2\)[/tex].
- Subtract:
[tex]\((-9x^3 + 4x) - (-9x^3 - 15x^2) = 15x^2 + 4x\)[/tex].
Step 5: Bring down the next term
Bring down [tex]\(-6\)[/tex] to get [tex]\(15x^2 + 4x - 6\)[/tex].
Step 6: Continue the process
- Divide [tex]\(15x^2\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5x\)[/tex].
- Multiply the divisor by [tex]\(5x\)[/tex] giving [tex]\(15x^2 + 25x\)[/tex].
- Subtract:
[tex]\((15x^2 + 4x - 6) - (15x^2 + 25x) = -21x - 6\)[/tex].
Step 7: Last step
- Divide [tex]\(-21x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-7\)[/tex].
- Multiply the divisor by [tex]\(-7\)[/tex] giving [tex]\(-21x - 35\)[/tex].
- Subtract:
[tex]\((-21x - 6) - (-21x - 35) = 29\)[/tex].
The quotient from this division is [tex]\(x^3 - 3x^2 + 5x - 7\)[/tex] with a remainder of [tex]\(29\)[/tex].
### Problem 20: Divide [tex]\(8x^3 + 10x^2 + 7x + 2\)[/tex] by [tex]\(2x + 1\)[/tex]
Step 1: Divide the first term
Divide [tex]\(8x^3\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(4x^2\)[/tex].
Step 2: Multiply and subtract
Multiply [tex]\(2x + 1\)[/tex] by [tex]\(4x^2\)[/tex] to get [tex]\(8x^3 + 4x^2\)[/tex].
Subtract this from the original to get:
[tex]\((8x^3 + 10x^2) - (8x^3 + 4x^2) = 6x^2\)[/tex].
Step 3: Bring down the next term
Bring down [tex]\(7x\)[/tex] to make [tex]\(6x^2 + 7x\)[/tex].
Step 4: Repeat the process
- Divide [tex]\(6x^2\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(3x\)[/tex].
- Multiply the divisor by [tex]\(3x\)[/tex] to get [tex]\(6x^2 + 3x\)[/tex].
- Subtract:
[tex]\((6x^2 + 7x) - (6x^2 + 3x) = 4x\)[/tex].
Step 5: Bring down the next term
Bring down [tex]\(2\)[/tex] to get [tex]\(4x + 2\)[/tex].
Step 6: Continue the process
- Divide [tex]\(4x\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(2\)[/tex].
- Multiply the divisor by [tex]\(2\)[/tex] to get [tex]\(4x + 2\)[/tex].
- Subtract:
[tex]\((4x + 2) - (4x + 2) = 0\)[/tex].
The quotient from this division is [tex]\(4x^2 + 3x + 2\)[/tex] with a remainder of [tex]\(0\)[/tex].
I hope this breakdown helps you understand polynomial long division better! If you have more questions, feel free to ask.
### Problem 19: Divide [tex]\(3x^4 - 4x^3 + 4x - 6\)[/tex] by [tex]\(3x + 5\)[/tex]
Step 1: Divide the first term
Take the first term of the dividend, [tex]\(3x^4\)[/tex], and divide it by the first term of the divisor, [tex]\(3x\)[/tex]. The result is [tex]\(x^3\)[/tex].
Step 2: Multiply and subtract
Multiply the entire divisor [tex]\(3x + 5\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(3x^4 + 5x^3\)[/tex].
Subtract this from the original dividend:
[tex]\((3x^4 - 4x^3) - (3x^4 + 5x^3) = -9x^3\)[/tex].
Step 3: Bring down the next term
Bring down the next term from the dividend, which is [tex]\(4x\)[/tex], to make [tex]\(-9x^3 + 4x\)[/tex].
Step 4: Repeat the process
- Divide [tex]\(-9x^3\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-3x^2\)[/tex].
- Multiply the divisor by [tex]\(-3x^2\)[/tex] giving [tex]\(-9x^3 - 15x^2\)[/tex].
- Subtract:
[tex]\((-9x^3 + 4x) - (-9x^3 - 15x^2) = 15x^2 + 4x\)[/tex].
Step 5: Bring down the next term
Bring down [tex]\(-6\)[/tex] to get [tex]\(15x^2 + 4x - 6\)[/tex].
Step 6: Continue the process
- Divide [tex]\(15x^2\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(5x\)[/tex].
- Multiply the divisor by [tex]\(5x\)[/tex] giving [tex]\(15x^2 + 25x\)[/tex].
- Subtract:
[tex]\((15x^2 + 4x - 6) - (15x^2 + 25x) = -21x - 6\)[/tex].
Step 7: Last step
- Divide [tex]\(-21x\)[/tex] by [tex]\(3x\)[/tex] to get [tex]\(-7\)[/tex].
- Multiply the divisor by [tex]\(-7\)[/tex] giving [tex]\(-21x - 35\)[/tex].
- Subtract:
[tex]\((-21x - 6) - (-21x - 35) = 29\)[/tex].
The quotient from this division is [tex]\(x^3 - 3x^2 + 5x - 7\)[/tex] with a remainder of [tex]\(29\)[/tex].
### Problem 20: Divide [tex]\(8x^3 + 10x^2 + 7x + 2\)[/tex] by [tex]\(2x + 1\)[/tex]
Step 1: Divide the first term
Divide [tex]\(8x^3\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(4x^2\)[/tex].
Step 2: Multiply and subtract
Multiply [tex]\(2x + 1\)[/tex] by [tex]\(4x^2\)[/tex] to get [tex]\(8x^3 + 4x^2\)[/tex].
Subtract this from the original to get:
[tex]\((8x^3 + 10x^2) - (8x^3 + 4x^2) = 6x^2\)[/tex].
Step 3: Bring down the next term
Bring down [tex]\(7x\)[/tex] to make [tex]\(6x^2 + 7x\)[/tex].
Step 4: Repeat the process
- Divide [tex]\(6x^2\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(3x\)[/tex].
- Multiply the divisor by [tex]\(3x\)[/tex] to get [tex]\(6x^2 + 3x\)[/tex].
- Subtract:
[tex]\((6x^2 + 7x) - (6x^2 + 3x) = 4x\)[/tex].
Step 5: Bring down the next term
Bring down [tex]\(2\)[/tex] to get [tex]\(4x + 2\)[/tex].
Step 6: Continue the process
- Divide [tex]\(4x\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(2\)[/tex].
- Multiply the divisor by [tex]\(2\)[/tex] to get [tex]\(4x + 2\)[/tex].
- Subtract:
[tex]\((4x + 2) - (4x + 2) = 0\)[/tex].
The quotient from this division is [tex]\(4x^2 + 3x + 2\)[/tex] with a remainder of [tex]\(0\)[/tex].
I hope this breakdown helps you understand polynomial long division better! If you have more questions, feel free to ask.
