Answer :
To find the quotient and remainder when dividing the polynomial [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex], we can use polynomial long division. Here's how it works, step-by-step:
1. Set up the division: Write [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] (note the zero placeholder for [tex]\(x^2\)[/tex]) under the division symbol, and [tex]\(x - 3\)[/tex] outside.
2. Divide the leading term: Divide the leading term of the dividend, [tex]\(3x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(3x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(3x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] to get [tex]\(3x^4 - 9x^3\)[/tex]. Subtract this product from the current 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: Bring down the next term in the dividend (if not already brought down) and repeat the process:
- Divide the new leading term, [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]\(21x^2 + 7x - 4\)[/tex].
5. Continue the division:
- 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]\(70x - 4\)[/tex].
6. Final step:
- 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 the remainder: [tex]\(206\)[/tex].
Finally, the quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex], and the remainder is [tex]\(206\)[/tex].
So, the answer is:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: [tex]\(206\)[/tex]
Select the correct choice:
- The correct answer is A: [tex]\(3x^3 + 7x^2 + 21x + 70\, ; \,206\)[/tex]
1. Set up the division: Write [tex]\(3x^4 - 2x^3 + 0x^2 + 7x - 4\)[/tex] (note the zero placeholder for [tex]\(x^2\)[/tex]) under the division symbol, and [tex]\(x - 3\)[/tex] outside.
2. Divide the leading term: Divide the leading term of the dividend, [tex]\(3x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives [tex]\(3x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(3x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] to get [tex]\(3x^4 - 9x^3\)[/tex]. Subtract this product from the current 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: Bring down the next term in the dividend (if not already brought down) and repeat the process:
- Divide the new leading term, [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]\(21x^2 + 7x - 4\)[/tex].
5. Continue the division:
- 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]\(70x - 4\)[/tex].
6. Final step:
- 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 the remainder: [tex]\(206\)[/tex].
Finally, the quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex], and the remainder is [tex]\(206\)[/tex].
So, the answer is:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: [tex]\(206\)[/tex]
Select the correct choice:
- The correct answer is A: [tex]\(3x^3 + 7x^2 + 21x + 70\, ; \,206\)[/tex]
