Answer :
To find the quotient and remainder of [tex]\((3x^4 - 2x^3 + 7x - 4) \div (x - 3)\)[/tex], we can use polynomial long division. Here's how you can perform the division step-by-step:
1. Set up the division:
- The dividend is [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex].
- The divisor is [tex]\(x - 3\)[/tex].
2. Divide the first term:
- Divide the leading term of the dividend [tex]\(3x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. The result is [tex]\(3x^3\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(3x^3\)[/tex], which gives [tex]\(3x^4 - 9x^3\)[/tex].
3. Subtract:
- Subtract [tex]\(3x^4 - 9x^3\)[/tex] from the dividend:
[tex]\[
(3x^4 - 2x^3) - (3x^4 - 9x^3) = 7x^3
\][/tex]
- Bring down the next term to get a new dividend: [tex]\(7x^3 + 0x^2\)[/tex].
4. Repeat the process:
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(7x^2\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(7x^2\)[/tex], which gives [tex]\(7x^3 - 21x^2\)[/tex].
- Subtract:
[tex]\[
(7x^3 + 0x^2) - (7x^3 - 21x^2) = 21x^2
\][/tex]
- Bring down the next term: [tex]\(21x^2 + 7x\)[/tex].
5. Continue dividing:
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(21x\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(21x\)[/tex], resulting in [tex]\(21x^2 - 63x\)[/tex].
- Subtract:
[tex]\[
(21x^2 + 7x) - (21x^2 - 63x) = 70x
\][/tex]
- Bring down the next term: [tex]\(70x - 4\)[/tex].
6. Final division step:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(70\)[/tex], resulting in [tex]\(70x - 210\)[/tex].
- Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
Now, the division gives us:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: 206
Therefore, the answer is option A:
[tex]\(3x^3 + 7x^2 + 21x + 70; 206\)[/tex]
1. Set up the division:
- The dividend is [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex].
- The divisor is [tex]\(x - 3\)[/tex].
2. Divide the first term:
- Divide the leading term of the dividend [tex]\(3x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. The result is [tex]\(3x^3\)[/tex].
- Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(3x^3\)[/tex], which gives [tex]\(3x^4 - 9x^3\)[/tex].
3. Subtract:
- Subtract [tex]\(3x^4 - 9x^3\)[/tex] from the dividend:
[tex]\[
(3x^4 - 2x^3) - (3x^4 - 9x^3) = 7x^3
\][/tex]
- Bring down the next term to get a new dividend: [tex]\(7x^3 + 0x^2\)[/tex].
4. Repeat the process:
- Divide [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(7x^2\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(7x^2\)[/tex], which gives [tex]\(7x^3 - 21x^2\)[/tex].
- Subtract:
[tex]\[
(7x^3 + 0x^2) - (7x^3 - 21x^2) = 21x^2
\][/tex]
- Bring down the next term: [tex]\(21x^2 + 7x\)[/tex].
5. Continue dividing:
- Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(21x\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(21x\)[/tex], resulting in [tex]\(21x^2 - 63x\)[/tex].
- Subtract:
[tex]\[
(21x^2 + 7x) - (21x^2 - 63x) = 70x
\][/tex]
- Bring down the next term: [tex]\(70x - 4\)[/tex].
6. Final division step:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply the divisor [tex]\(x - 3\)[/tex] by [tex]\(70\)[/tex], resulting in [tex]\(70x - 210\)[/tex].
- Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
Now, the division gives us:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: 206
Therefore, the answer is option A:
[tex]\(3x^3 + 7x^2 + 21x + 70; 206\)[/tex]
