Answer :
Sure, let's go through the division of the polynomials step by step.
We are dividing the polynomial [tex]\(6x^3 - 19x^2 + 27x - 15\)[/tex] by [tex]\(3x - 5\)[/tex].
Step 1: Setup the division
We'll use polynomial long division, which is similar to numerical long division. Place [tex]\(6x^3 - 19x^2 + 27x - 15\)[/tex] (the dividend) under the division symbol and [tex]\(3x - 5\)[/tex] (the divisor) to the left of it.
Step 2: Divide the first terms
- Divide the first term of the dividend, [tex]\(6x^3\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex].
- This gives us: [tex]\(\frac{6x^3}{3x} = 2x^2\)[/tex].
Step 3: Multiply and subtract
- Multiply [tex]\(2x^2\)[/tex] by the entire divisor [tex]\(3x - 5\)[/tex] to get [tex]\(2x^2 \cdot (3x - 5) = 6x^3 - 10x^2\)[/tex].
- Subtract this from the current dividend:
[tex]\[ (6x^3 - 19x^2 + 27x - 15) - (6x^3 - 10x^2) = -9x^2 + 27x - 15 \][/tex].
Step 4: Repeat the process
- Divide the first term of the new dividend, [tex]\(-9x^2\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[\frac{-9x^2}{3x} = -3x\][/tex].
- Multiply [tex]\(-3x\)[/tex] by [tex]\(3x - 5\)[/tex] to get [tex]\(-9x^2 + 15x\)[/tex].
- Subtract this from [tex]\(-9x^2 + 27x - 15\)[/tex]:
[tex]\[ (-9x^2 + 27x - 15) - (-9x^2 + 15x) = 12x - 15 \][/tex].
Step 5: Continue the process
- Divide the first term of the result, [tex]\(12x\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[\frac{12x}{3x} = 4\][/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(3x - 5\)[/tex] to get [tex]\(12x - 20\)[/tex].
- Subtract this from [tex]\(12x - 15\)[/tex]:
[tex]\[ (12x - 15) - (12x - 20) = 5 \][/tex].
Conclusion
The quotient of the division is [tex]\(2x^2 - 3x + 4\)[/tex] and the remainder is [tex]\(5\)[/tex].
So,
[tex]\[ \frac{6x^3 - 19x^2 + 27x - 15}{3x - 5} = 2x^2 - 3x + 4 \][/tex] with a remainder of [tex]\(5\)[/tex].
We are dividing the polynomial [tex]\(6x^3 - 19x^2 + 27x - 15\)[/tex] by [tex]\(3x - 5\)[/tex].
Step 1: Setup the division
We'll use polynomial long division, which is similar to numerical long division. Place [tex]\(6x^3 - 19x^2 + 27x - 15\)[/tex] (the dividend) under the division symbol and [tex]\(3x - 5\)[/tex] (the divisor) to the left of it.
Step 2: Divide the first terms
- Divide the first term of the dividend, [tex]\(6x^3\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex].
- This gives us: [tex]\(\frac{6x^3}{3x} = 2x^2\)[/tex].
Step 3: Multiply and subtract
- Multiply [tex]\(2x^2\)[/tex] by the entire divisor [tex]\(3x - 5\)[/tex] to get [tex]\(2x^2 \cdot (3x - 5) = 6x^3 - 10x^2\)[/tex].
- Subtract this from the current dividend:
[tex]\[ (6x^3 - 19x^2 + 27x - 15) - (6x^3 - 10x^2) = -9x^2 + 27x - 15 \][/tex].
Step 4: Repeat the process
- Divide the first term of the new dividend, [tex]\(-9x^2\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[\frac{-9x^2}{3x} = -3x\][/tex].
- Multiply [tex]\(-3x\)[/tex] by [tex]\(3x - 5\)[/tex] to get [tex]\(-9x^2 + 15x\)[/tex].
- Subtract this from [tex]\(-9x^2 + 27x - 15\)[/tex]:
[tex]\[ (-9x^2 + 27x - 15) - (-9x^2 + 15x) = 12x - 15 \][/tex].
Step 5: Continue the process
- Divide the first term of the result, [tex]\(12x\)[/tex], by the first term of the divisor, [tex]\(3x\)[/tex]:
[tex]\[\frac{12x}{3x} = 4\][/tex].
- Multiply [tex]\(4\)[/tex] by [tex]\(3x - 5\)[/tex] to get [tex]\(12x - 20\)[/tex].
- Subtract this from [tex]\(12x - 15\)[/tex]:
[tex]\[ (12x - 15) - (12x - 20) = 5 \][/tex].
Conclusion
The quotient of the division is [tex]\(2x^2 - 3x + 4\)[/tex] and the remainder is [tex]\(5\)[/tex].
So,
[tex]\[ \frac{6x^3 - 19x^2 + 27x - 15}{3x - 5} = 2x^2 - 3x + 4 \][/tex] with a remainder of [tex]\(5\)[/tex].
