Answer :
To find the quotient of the polynomial [tex]\((5x^5 + 90x^2 - 135x)\)[/tex] divided by [tex]\((x + 3)\)[/tex], we can use long division. I'll take you through the process step-by-step:
1. Set up the division.
We will divide [tex]\(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0\)[/tex] (adding missing terms) by [tex]\(x + 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor.
Divide [tex]\(5x^5\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(5x^4\)[/tex].
3. Multiply and subtract.
- Multiply [tex]\(5x^4\)[/tex] by [tex]\(x + 3\)[/tex], giving [tex]\(5x^5 + 15x^4\)[/tex].
- Subtract [tex]\((5x^5 + 15x^4)\)[/tex] from the original polynomial:
[tex]\[
(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0) - (5x^5 + 15x^4) = -15x^4 + 0x^3 + 90x^2 - 135x
\][/tex]
4. Repeat the process.
- Divide [tex]\(-15x^4\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-15x^3\)[/tex].
- Multiply [tex]\(-15x^3\)[/tex] by [tex]\(x + 3\)[/tex], giving [tex]\(-15x^4 - 45x^3\)[/tex].
- Subtract:
[tex]\[
(-15x^4 + 0x^3 + 90x^2 - 135x) - (-15x^4 - 45x^3) = 45x^3 + 90x^2 - 135x
\][/tex]
5. Continue the process.
- Divide [tex]\(45x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(45x^2\)[/tex].
- Multiply [tex]\(45x^2\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(45x^3 + 135x^2\)[/tex].
- Subtract:
[tex]\[
(45x^3 + 90x^2 - 135x) - (45x^3 + 135x^2) = -45x^2 - 135x
\][/tex]
6. Repeat one last time.
- Divide [tex]\(-45x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-45x\)[/tex].
- Multiply [tex]\(-45x\)[/tex] by [tex]\(x + 3\)[/tex], which is [tex]\(-45x^2 - 135x\)[/tex].
- Subtract:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
7. Conclusion:
After completing the division, we get the quotient: [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex] and a remainder of 0. Therefore, the answer is:
D. [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex]
1. Set up the division.
We will divide [tex]\(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0\)[/tex] (adding missing terms) by [tex]\(x + 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor.
Divide [tex]\(5x^5\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(5x^4\)[/tex].
3. Multiply and subtract.
- Multiply [tex]\(5x^4\)[/tex] by [tex]\(x + 3\)[/tex], giving [tex]\(5x^5 + 15x^4\)[/tex].
- Subtract [tex]\((5x^5 + 15x^4)\)[/tex] from the original polynomial:
[tex]\[
(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0) - (5x^5 + 15x^4) = -15x^4 + 0x^3 + 90x^2 - 135x
\][/tex]
4. Repeat the process.
- Divide [tex]\(-15x^4\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-15x^3\)[/tex].
- Multiply [tex]\(-15x^3\)[/tex] by [tex]\(x + 3\)[/tex], giving [tex]\(-15x^4 - 45x^3\)[/tex].
- Subtract:
[tex]\[
(-15x^4 + 0x^3 + 90x^2 - 135x) - (-15x^4 - 45x^3) = 45x^3 + 90x^2 - 135x
\][/tex]
5. Continue the process.
- Divide [tex]\(45x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(45x^2\)[/tex].
- Multiply [tex]\(45x^2\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(45x^3 + 135x^2\)[/tex].
- Subtract:
[tex]\[
(45x^3 + 90x^2 - 135x) - (45x^3 + 135x^2) = -45x^2 - 135x
\][/tex]
6. Repeat one last time.
- Divide [tex]\(-45x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-45x\)[/tex].
- Multiply [tex]\(-45x\)[/tex] by [tex]\(x + 3\)[/tex], which is [tex]\(-45x^2 - 135x\)[/tex].
- Subtract:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
7. Conclusion:
After completing the division, we get the quotient: [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex] and a remainder of 0. Therefore, the answer is:
D. [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex]
