Answer :
To solve the problem of dividing [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex], we are essentially performing polynomial long division. Here's a step-by-step process:
1. Set up the division: We have the dividend [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] (note the [tex]\(0x^2\)[/tex] term to keep the places aligned) and divisor [tex]\(x - 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(3x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex], which gives [tex]\(3x^3\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 3\)[/tex] by the result from step 2, [tex]\(3x^3\)[/tex]:
[tex]\[
(3x^3)(x - 3) = 3x^4 - 9x^3
\][/tex]
Now subtract this from the original dividend:
[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 the new leading term [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(7x^2\)[/tex].
- Multiply the divisor by [tex]\(7x^2\)[/tex]:
[tex]\[
(7x^2)(x - 3) = 7x^3 - 21x^2
\][/tex]
- Subtract:
[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 the divisor by [tex]\(21x\)[/tex]:
[tex]\[
(21x)(x - 3) = 21x^2 - 63x
\][/tex]
- Subtract:
[tex]\[
(21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4
\][/tex]
6. Final division:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply the divisor by [tex]\(70\)[/tex]:
[tex]\[
(70)(x - 3) = 70x - 210
\][/tex]
- Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is [tex]\(206\)[/tex].
Therefore, the correct answer is A: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]; 206.
1. Set up the division: We have the dividend [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] (note the [tex]\(0x^2\)[/tex] term to keep the places aligned) and divisor [tex]\(x - 3\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend, [tex]\(3x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex], which gives [tex]\(3x^3\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x - 3\)[/tex] by the result from step 2, [tex]\(3x^3\)[/tex]:
[tex]\[
(3x^3)(x - 3) = 3x^4 - 9x^3
\][/tex]
Now subtract this from the original dividend:
[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 the new leading term [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(7x^2\)[/tex].
- Multiply the divisor by [tex]\(7x^2\)[/tex]:
[tex]\[
(7x^2)(x - 3) = 7x^3 - 21x^2
\][/tex]
- Subtract:
[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 the divisor by [tex]\(21x\)[/tex]:
[tex]\[
(21x)(x - 3) = 21x^2 - 63x
\][/tex]
- Subtract:
[tex]\[
(21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4
\][/tex]
6. Final division:
- Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(70\)[/tex].
- Multiply the divisor by [tex]\(70\)[/tex]:
[tex]\[
(70)(x - 3) = 70x - 210
\][/tex]
- Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is [tex]\(206\)[/tex].
Therefore, the correct answer is A: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]; 206.
