Answer :
Certainly! Let's go through the process of using long division to divide the polynomial [tex]\(5x^5 + 90x^2 - 135x\)[/tex] by [tex]\(x + 3\)[/tex].
### Step-by-Step Solution:
1. Set up the division:
Write the dividend, [tex]\(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0\)[/tex], under the division bar and the divisor [tex]\(x + 3\)[/tex] outside.
2. Divide the highest degree term:
- Divide the first term of the dividend, [tex]\(5x^5\)[/tex], by the first term of the divisor, [tex]\(x\)[/tex].
- [tex]\(\frac{5x^5}{x} = 5x^4\)[/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 current dividend terms:
[tex]\[
(5x^5 + 0x^4) - (5x^5 + 15x^4) = -15x^4
\][/tex]
- Bring down the next term, [tex]\(0x^3\)[/tex], from the original dividend, getting [tex]\(-15x^4 + 0x^3\)[/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]\[
(-15x^4 + 0x^3) - (-15x^4 - 45x^3) = 45x^3
\][/tex]
- Bring down the next term, [tex]\(90x^2\)[/tex], getting [tex]\(45x^3 + 90x^2\)[/tex].
5. Continue dividing:
- 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]\[
(45x^3 + 90x^2) - (45x^3 + 135x^2) = -45x^2
\][/tex]
- Bring down the [tex]\(-135x\)[/tex], resulting in [tex]\(-45x^2 - 135x\)[/tex].
6. Repeat the division:
- Divide [tex]\(-45x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-45x\)[/tex].
- Multiply [tex]\(-45x\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(-45x^2 - 135x\)[/tex].
- Subtract:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
The remainder is 0, indicating that [tex]\(x + 3\)[/tex] divides the polynomial evenly.
### Conclusion:
The quotient is [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex].
The correct answer is A: [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex].
### Step-by-Step Solution:
1. Set up the division:
Write the dividend, [tex]\(5x^5 + 0x^4 + 0x^3 + 90x^2 - 135x + 0\)[/tex], under the division bar and the divisor [tex]\(x + 3\)[/tex] outside.
2. Divide the highest degree term:
- Divide the first term of the dividend, [tex]\(5x^5\)[/tex], by the first term of the divisor, [tex]\(x\)[/tex].
- [tex]\(\frac{5x^5}{x} = 5x^4\)[/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 current dividend terms:
[tex]\[
(5x^5 + 0x^4) - (5x^5 + 15x^4) = -15x^4
\][/tex]
- Bring down the next term, [tex]\(0x^3\)[/tex], from the original dividend, getting [tex]\(-15x^4 + 0x^3\)[/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]\[
(-15x^4 + 0x^3) - (-15x^4 - 45x^3) = 45x^3
\][/tex]
- Bring down the next term, [tex]\(90x^2\)[/tex], getting [tex]\(45x^3 + 90x^2\)[/tex].
5. Continue dividing:
- 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]\[
(45x^3 + 90x^2) - (45x^3 + 135x^2) = -45x^2
\][/tex]
- Bring down the [tex]\(-135x\)[/tex], resulting in [tex]\(-45x^2 - 135x\)[/tex].
6. Repeat the division:
- Divide [tex]\(-45x^2\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-45x\)[/tex].
- Multiply [tex]\(-45x\)[/tex] by [tex]\(x + 3\)[/tex], resulting in [tex]\(-45x^2 - 135x\)[/tex].
- Subtract:
[tex]\[
(-45x^2 - 135x) - (-45x^2 - 135x) = 0
\][/tex]
The remainder is 0, indicating that [tex]\(x + 3\)[/tex] divides the polynomial evenly.
### Conclusion:
The quotient is [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex].
The correct answer is A: [tex]\(5x^4 - 15x^3 + 45x^2 - 45x\)[/tex].
