Answer :
To solve the problem of dividing [tex]\(x^4 + 7\)[/tex] by [tex]\(x - 3\)[/tex], we'll go through the process of polynomial long division to find the quotient and remainder.
1. Set up the division: Write [tex]\(x^4 + 7\)[/tex] as the dividend and [tex]\(x - 3\)[/tex] as the divisor. You'll need to consider each term of the dividend separately, aligning with powers of [tex]\(x\)[/tex].
2. Divide the first term: Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives us [tex]\(x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] to get [tex]\(x^4 - 3x^3\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 0x^3 + 0x^2 + 0x + 7) - (x^4 - 3x^3) = 3x^3 + 0x^2 + 0x + 7
\][/tex]
4. Bring down the next term: The next term to consider is [tex]\(3x^3\)[/tex]. Divide [tex]\(3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(3x^2\)[/tex].
5. Multiply and subtract: Multiply [tex]\(3x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(3x^3 - 9x^2\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(3x^3 + 0x^2 + 0x + 7) - (3x^3 - 9x^2) = 9x^2 + 0x + 7
\][/tex]
6. Bring down the next term: Divide [tex]\(9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(9x\)[/tex].
7. Multiply and subtract: Multiply [tex]\(9x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(9x^2 - 27x\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(9x^2 + 0x + 7) - (9x^2 - 27x) = 27x + 7
\][/tex]
8. Bring down the next term: Divide [tex]\(27x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(27\)[/tex].
9. Multiply and subtract: Multiply [tex]\(27\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(27x - 81\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(27x + 7) - (27x - 81) = 88
\][/tex]
10. Result: The polynomial division is complete, and we are left with a remainder of 88. Therefore, the quotient is [tex]\(x^3 + 3x^2 + 9x + 27\)[/tex] and the remainder is 88.
The correct step-by-step division results in the quotient [tex]\(x^3 + 3x^2 + 9x + 27\)[/tex] with a remainder of 88, matching the first set of conditions in the given options:
[tex]\[ x^3 + 3x^2 + 9x + 27 \quad \text{R } 88 \][/tex]
So, the solution is:
[tex]\[ x^3 + 3x^2 + 9x + 27 \quad \text{R } 88 \][/tex]
1. Set up the division: Write [tex]\(x^4 + 7\)[/tex] as the dividend and [tex]\(x - 3\)[/tex] as the divisor. You'll need to consider each term of the dividend separately, aligning with powers of [tex]\(x\)[/tex].
2. Divide the first term: Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x\)[/tex]. This gives us [tex]\(x^3\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex] to get [tex]\(x^4 - 3x^3\)[/tex]. Subtract this from the original dividend:
[tex]\[
(x^4 + 0x^3 + 0x^2 + 0x + 7) - (x^4 - 3x^3) = 3x^3 + 0x^2 + 0x + 7
\][/tex]
4. Bring down the next term: The next term to consider is [tex]\(3x^3\)[/tex]. Divide [tex]\(3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(3x^2\)[/tex].
5. Multiply and subtract: Multiply [tex]\(3x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(3x^3 - 9x^2\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(3x^3 + 0x^2 + 0x + 7) - (3x^3 - 9x^2) = 9x^2 + 0x + 7
\][/tex]
6. Bring down the next term: Divide [tex]\(9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(9x\)[/tex].
7. Multiply and subtract: Multiply [tex]\(9x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(9x^2 - 27x\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(9x^2 + 0x + 7) - (9x^2 - 27x) = 27x + 7
\][/tex]
8. Bring down the next term: Divide [tex]\(27x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(27\)[/tex].
9. Multiply and subtract: Multiply [tex]\(27\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(27x - 81\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(27x + 7) - (27x - 81) = 88
\][/tex]
10. Result: The polynomial division is complete, and we are left with a remainder of 88. Therefore, the quotient is [tex]\(x^3 + 3x^2 + 9x + 27\)[/tex] and the remainder is 88.
The correct step-by-step division results in the quotient [tex]\(x^3 + 3x^2 + 9x + 27\)[/tex] with a remainder of 88, matching the first set of conditions in the given options:
[tex]\[ x^3 + 3x^2 + 9x + 27 \quad \text{R } 88 \][/tex]
So, the solution is:
[tex]\[ x^3 + 3x^2 + 9x + 27 \quad \text{R } 88 \][/tex]
