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Answer :
Sure, let's solve the problem step-by-step using polynomial long division.
The expression we need to divide is [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex].
### Step 1: Divide the leading terms
- Leading term of the dividend: [tex]\(3x^4\)[/tex]
- Leading term of the divisor: [tex]\(x\)[/tex]
Divide [tex]\(3x^4\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(3x^3\)[/tex].
### Step 2: Multiply and subtract
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(3x^3\)[/tex]:
[tex]\((x - 3) \times 3x^3 = 3x^4 - 9x^3\)[/tex]
- Subtract this result from the original dividend:
[tex]\((3x^4 - 2x^3 + 0x^2 + 7x - 4) - (3x^4 - 9x^3)\)[/tex]
This yields a new polynomial: [tex]\(7x^3 + 0x^2 + 7x - 4\)[/tex].
### Step 3: Repeat the process
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(7x^2\)[/tex].
- Multiply [tex]\((x - 3) \times 7x^2 = 7x^3 - 21x^2\)[/tex].
- Subtract:
[tex]\((7x^3 + 0x^2 + 7x - 4) - (7x^3 - 21x^2)\)[/tex]
New polynomial: [tex]\(21x^2 + 7x - 4\)[/tex].
### Step 4: Repeat again
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(21x\)[/tex].
- Multiply [tex]\((x - 3) \times 21x = 21x^2 - 63x\)[/tex].
- Subtract:
[tex]\((21x^2 + 7x - 4) - (21x^2 - 63x)\)[/tex]
New polynomial: [tex]\(70x - 4\)[/tex].
### Step 5: Final repeat
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(70\)[/tex].
- Multiply [tex]\((x - 3) \times 70 = 70x - 210\)[/tex].
- Subtract:
[tex]\((70x - 4) - (70x - 210)\)[/tex]
New polynomial: [tex]\(206\)[/tex].
### Conclusion
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex], and the remainder is [tex]\(-206\)[/tex].
So, the correct answer is:
b. [tex]\(3x^3 + 7x^2 + 21x + 70 ; -206\)[/tex]
The expression we need to divide is [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex].
### Step 1: Divide the leading terms
- Leading term of the dividend: [tex]\(3x^4\)[/tex]
- Leading term of the divisor: [tex]\(x\)[/tex]
Divide [tex]\(3x^4\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(3x^3\)[/tex].
### Step 2: Multiply and subtract
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(3x^3\)[/tex]:
[tex]\((x - 3) \times 3x^3 = 3x^4 - 9x^3\)[/tex]
- Subtract this result from the original dividend:
[tex]\((3x^4 - 2x^3 + 0x^2 + 7x - 4) - (3x^4 - 9x^3)\)[/tex]
This yields a new polynomial: [tex]\(7x^3 + 0x^2 + 7x - 4\)[/tex].
### Step 3: Repeat the process
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(7x^2\)[/tex].
- Multiply [tex]\((x - 3) \times 7x^2 = 7x^3 - 21x^2\)[/tex].
- Subtract:
[tex]\((7x^3 + 0x^2 + 7x - 4) - (7x^3 - 21x^2)\)[/tex]
New polynomial: [tex]\(21x^2 + 7x - 4\)[/tex].
### Step 4: Repeat again
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(21x\)[/tex].
- Multiply [tex]\((x - 3) \times 21x = 21x^2 - 63x\)[/tex].
- Subtract:
[tex]\((21x^2 + 7x - 4) - (21x^2 - 63x)\)[/tex]
New polynomial: [tex]\(70x - 4\)[/tex].
### Step 5: Final repeat
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(70\)[/tex].
- Multiply [tex]\((x - 3) \times 70 = 70x - 210\)[/tex].
- Subtract:
[tex]\((70x - 4) - (70x - 210)\)[/tex]
New polynomial: [tex]\(206\)[/tex].
### Conclusion
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex], and the remainder is [tex]\(-206\)[/tex].
So, the correct answer is:
b. [tex]\(3x^3 + 7x^2 + 21x + 70 ; -206\)[/tex]
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