Answer :
To solve the division of the polynomial [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex] by [tex]\(x-3\)[/tex], we use polynomial long division. The goal is to find both the quotient and the remainder.
### Step-by-Step Solution:
1. Setup the division:
- Dividend: [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex]
- Divisor: [tex]\(x - 3\)[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
- Divide [tex]\(3x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3x^3\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(3x^3\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(3x^4 - 9x^3\)[/tex].
- Subtract [tex]\(3x^4 - 9x^3\)[/tex] from the current dividend to get:
[tex]\[
(3x^4 - 2x^3 + 0x^2 + 7x - 4) - (3x^4 - 9x^3) = 7x^3 + 0x^2 + 7x - 4
\][/tex]
4. Repeat the process:
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(7x^2\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(7x^3 - 21x^2\)[/tex].
- Subtract to get: [tex]\((7x^3 + 0x^2 + 7x - 4) - (7x^3 - 21x^2) = 21x^2 + 7x - 4\)[/tex].
5. Continue dividing:
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(21x\)[/tex].
- Multiply [tex]\(21x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(21x^2 - 63x\)[/tex].
- Subtract to get: [tex]\((21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4\)[/tex].
6. Divide 70x by x:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply [tex]\(70\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(70x - 210\)[/tex].
- Subtract to get: [tex]\(70x - 4 - (70x - 210) = 206\)[/tex].
7. Final results:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: [tex]\(206\)[/tex]
By performing this division, we identify the quotient and remainder as follows:
- The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex].
- The remainder is [tex]\(206\)[/tex].
Therefore, the correct answer is Option A:
[tex]\[ \text{Quotient: } 3x^3 + 7x^2 + 21x + 70; \text{ Remainder: } 206. \][/tex]
### Step-by-Step Solution:
1. Setup the division:
- Dividend: [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex]
- Divisor: [tex]\(x - 3\)[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
- Divide [tex]\(3x^4\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(3x^3\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(3x^3\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(3x^4 - 9x^3\)[/tex].
- Subtract [tex]\(3x^4 - 9x^3\)[/tex] from the current dividend to get:
[tex]\[
(3x^4 - 2x^3 + 0x^2 + 7x - 4) - (3x^4 - 9x^3) = 7x^3 + 0x^2 + 7x - 4
\][/tex]
4. Repeat the process:
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(7x^2\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(7x^3 - 21x^2\)[/tex].
- Subtract to get: [tex]\((7x^3 + 0x^2 + 7x - 4) - (7x^3 - 21x^2) = 21x^2 + 7x - 4\)[/tex].
5. Continue dividing:
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(21x\)[/tex].
- Multiply [tex]\(21x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(21x^2 - 63x\)[/tex].
- Subtract to get: [tex]\((21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4\)[/tex].
6. Divide 70x by x:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply [tex]\(70\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(70x - 210\)[/tex].
- Subtract to get: [tex]\(70x - 4 - (70x - 210) = 206\)[/tex].
7. Final results:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: [tex]\(206\)[/tex]
By performing this division, we identify the quotient and remainder as follows:
- The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex].
- The remainder is [tex]\(206\)[/tex].
Therefore, the correct answer is Option A:
[tex]\[ \text{Quotient: } 3x^3 + 7x^2 + 21x + 70; \text{ Remainder: } 206. \][/tex]
