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 use polynomial division.
### Step-by-step Solution:
1. Setup the division: We are dividing the polynomial [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex].
2. Perform synthetic division or long division:
- Synthetic division is a simplified form of polynomial division, where we only use the coefficients of the polynomials.
- For [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex], the coefficients are [tex]\([3, -2, 0, 7, -4]\)[/tex]. Note that there is no [tex]\(x^2\)[/tex] term, so its coefficient is 0.
3. Divide step-by-step:
- Start with the first coefficient (3). It's divided directly by the leading term of the divisor (1 for [tex]\(x\)[/tex]) to start forming the quotient.
- Multiply the divisor root (the number to subtract, in this case, 3) by the current term in the quotient, then subtract the result from the next coefficient in the dividend.
- Repeat these steps for each coefficient.
4. Obtained Quotient and Remainder:
- Quotient: The result of the division is the polynomial [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex].
- Remainder: The leftover value after division is 206.
Thus, the quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is 206.
Matching this with the given options, the correct answer is:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: 206
Therefore, the correct choice is the first one.
### Step-by-step Solution:
1. Setup the division: We are dividing the polynomial [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex] by [tex]\(x - 3\)[/tex].
2. Perform synthetic division or long division:
- Synthetic division is a simplified form of polynomial division, where we only use the coefficients of the polynomials.
- For [tex]\(3x^4 - 2x^3 + 7x - 4\)[/tex], the coefficients are [tex]\([3, -2, 0, 7, -4]\)[/tex]. Note that there is no [tex]\(x^2\)[/tex] term, so its coefficient is 0.
3. Divide step-by-step:
- Start with the first coefficient (3). It's divided directly by the leading term of the divisor (1 for [tex]\(x\)[/tex]) to start forming the quotient.
- Multiply the divisor root (the number to subtract, in this case, 3) by the current term in the quotient, then subtract the result from the next coefficient in the dividend.
- Repeat these steps for each coefficient.
4. Obtained Quotient and Remainder:
- Quotient: The result of the division is the polynomial [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex].
- Remainder: The leftover value after division is 206.
Thus, the quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is 206.
Matching this with the given options, the correct answer is:
- Quotient: [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex]
- Remainder: 206
Therefore, the correct choice is the first one.
