College

What are the quotient and remainder of [tex]\left(3x^4 - 2x^3 + 7x - 4\right) \div (x-3)[/tex]?

A. [tex]3x^3 + 7x^2 + 21x + 70 ; 206[/tex]
B. [tex]3x^3 + 7x^2 + 21x + 70 ; -206[/tex]
C. [tex]3x^4 + 7x^3 + 21x^2 + 70 ; 206[/tex]
D. [tex]3x^3 + 7x^2 + 21x + 70 ; 76[/tex]

Please select the best answer from the choices provided:
A, B, C, D

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]