Answer :
To solve the given expression, we need to perform polynomial division on [tex]\(9x^3 - 15x^2 + 27x\)[/tex] by [tex]\(3x\)[/tex].
### Step-by-Step Solution:
1. Identify the Dividend and Divisor:
- Dividend Polynomial: [tex]\(9x^3 - 15x^2 + 27x\)[/tex]
- Divisor Polynomial: [tex]\(3x\)[/tex]
2. Perform Polynomial Division:
- We divide the first term of the dividend by the first term of the divisor to find the leading term of the quotient.
- [tex]\(\frac{9x^3}{3x} = 3x^2\)[/tex]
- Multiply the entire divisor [tex]\(3x\)[/tex] by the obtained quotient term [tex]\(3x^2\)[/tex], which gives [tex]\(9x^3\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(9x^3 - 15x^2 + 27x) - (9x^3) = -15x^2 + 27x
\][/tex]
- Again, divide the new leading term [tex]\(-15x^2\)[/tex] by [tex]\(3x\)[/tex]:
- [tex]\(\frac{-15x^2}{3x} = -5x\)[/tex]
- Multiply the entire divisor [tex]\(3x\)[/tex] by [tex]\(-5x\)[/tex], resulting in [tex]\(-15x^2\)[/tex].
- Subtract from the current expression:
[tex]\[
(-15x^2 + 27x) - (-15x^2) = 27x
\][/tex]
- Now divide the term [tex]\(27x\)[/tex] by [tex]\(3x\)[/tex]:
- [tex]\(\frac{27x}{3x} = 9\)[/tex]
- Multiply the divisor by this term: [tex]\(3x \times 9 = 27x\)[/tex].
- Subtract to find the remainder:
[tex]\[
27x - 27x = 0
\][/tex]
3. Final Quotient and Remainder:
- Quotient: [tex]\(3x^2 - 5x + 9\)[/tex]
- Remainder: [tex]\(0\)[/tex]
The division results in a quotient of [tex]\(3x^2 - 5x + 9\)[/tex] with no remainder. This means the given dividend is completely divisible by the divisor.
### Step-by-Step Solution:
1. Identify the Dividend and Divisor:
- Dividend Polynomial: [tex]\(9x^3 - 15x^2 + 27x\)[/tex]
- Divisor Polynomial: [tex]\(3x\)[/tex]
2. Perform Polynomial Division:
- We divide the first term of the dividend by the first term of the divisor to find the leading term of the quotient.
- [tex]\(\frac{9x^3}{3x} = 3x^2\)[/tex]
- Multiply the entire divisor [tex]\(3x\)[/tex] by the obtained quotient term [tex]\(3x^2\)[/tex], which gives [tex]\(9x^3\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(9x^3 - 15x^2 + 27x) - (9x^3) = -15x^2 + 27x
\][/tex]
- Again, divide the new leading term [tex]\(-15x^2\)[/tex] by [tex]\(3x\)[/tex]:
- [tex]\(\frac{-15x^2}{3x} = -5x\)[/tex]
- Multiply the entire divisor [tex]\(3x\)[/tex] by [tex]\(-5x\)[/tex], resulting in [tex]\(-15x^2\)[/tex].
- Subtract from the current expression:
[tex]\[
(-15x^2 + 27x) - (-15x^2) = 27x
\][/tex]
- Now divide the term [tex]\(27x\)[/tex] by [tex]\(3x\)[/tex]:
- [tex]\(\frac{27x}{3x} = 9\)[/tex]
- Multiply the divisor by this term: [tex]\(3x \times 9 = 27x\)[/tex].
- Subtract to find the remainder:
[tex]\[
27x - 27x = 0
\][/tex]
3. Final Quotient and Remainder:
- Quotient: [tex]\(3x^2 - 5x + 9\)[/tex]
- Remainder: [tex]\(0\)[/tex]
The division results in a quotient of [tex]\(3x^2 - 5x + 9\)[/tex] with no remainder. This means the given dividend is completely divisible by the divisor.
