Answer :
Sure! Let's work through the problem step-by-step using synthetic division for each rational expression:
### Part (a)
[tex]\[ \frac{x^2 - 17x - 33}{x + 2} \][/tex]
Here, we are dividing [tex]\(x^2 - 17x - 33\)[/tex] by [tex]\(x + 2\)[/tex].
1. Set [tex]\(x + 2 = 0\)[/tex] which gives [tex]\(x = -2\)[/tex], the value we will use for synthetic division.
2. Coefficients of the polynomial [tex]\(x^2 - 17x - 33\)[/tex] are [tex]\( [1, -17, -33] \)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(1, -17, -33\)[/tex].
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply by [tex]\(-2\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(1 \cdot (-2) = -2\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-17 + (-2) = -19\)[/tex].
- Multiply by [tex]\(-2\)[/tex] again: [tex]\(-19 \cdot (-2) = 38\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-33 + 38 = 5\)[/tex].
The quotient is [tex]\(x - 19\)[/tex] and the remainder is [tex]\(5\)[/tex]. So:
[tex]\[ \frac{x^2 - 17x - 33}{x + 2} = (x - 19) + \frac{5}{x + 2} \][/tex]
### Part (b)
[tex]\[ \frac{2x^2 - 9x + 19}{x - 6} \][/tex]
Here, we are dividing [tex]\(2x^2 - 9x + 19\)[/tex] by [tex]\(x - 6\)[/tex].
1. Set [tex]\(x - 6 = 0\)[/tex] which gives [tex]\(x = 6\)[/tex].
2. Coefficients of the polynomial [tex]\(2x^2 - 9x + 19\)[/tex] are [tex]\([2, -9, 19]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(2, -9, 19\)[/tex].
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply by [tex]\(6\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(2 \cdot 6 = 12\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-9 + 12 = 3\)[/tex].
- Multiply by [tex]\(6\)[/tex] again: [tex]\(3 \cdot 6 = 18\)[/tex].
- Write the result under the next coefficient and add: [tex]\(19 + 18 = 37\)[/tex].
The quotient is [tex]\(2x + 3\)[/tex] and the remainder is [tex]\(37\)[/tex]. So:
[tex]\[ \frac{2x^2 - 9x + 19}{x - 6} = (2x + 3) + \frac{37}{x - 6} \][/tex]
### Part (c)
[tex]\[ \frac{x^3 - 2x^2 + 7x - 25}{x - 3} \][/tex]
Here, we are dividing [tex]\(x^3 - 2x^2 + 7x - 25\)[/tex] by [tex]\(x - 3\)[/tex].
1. Set [tex]\(x - 3 = 0\)[/tex] which gives [tex]\(x = 3\)[/tex].
2. Coefficients of the polynomial [tex]\(x^3 - 2x^2 + 7x - 25\)[/tex] are [tex]\([1, -2, 7, -25]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(1, -2, 7, -25\)[/tex].
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply by [tex]\(3\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(1 \cdot 3 = 3\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-2 + 3 = 1\)[/tex].
- Multiply by [tex]\(3\)[/tex] again: [tex]\(1 \cdot 3 = 3\)[/tex].
- Write the result under the next coefficient and add: [tex]\(7 + 3 = 10\)[/tex].
- Multiply by [tex]\(3\)[/tex] again: [tex]\(10 \cdot 3 = 30\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-25 + 30 = 5\)[/tex].
The quotient is [tex]\(x^2 + x + 10\)[/tex] and the remainder is [tex]\(5\)[/tex]. So:
[tex]\[ \frac{x^3 - 2x^2 + 7x - 25}{x - 3} = (x^2 + x + 10) + \frac{5}{x - 3} \][/tex]
### Part (d)
[tex]\[ \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} \][/tex]
Here, we are dividing [tex]\(2x^3 + 13x^2 + 20x - 12\)[/tex] by [tex]\(x + 4\)[/tex].
1. Set [tex]\(x + 4 = 0\)[/tex] which gives [tex]\(x = -4\)[/tex].
2. Coefficients of the polynomial [tex]\(2x^3 + 13x^2 + 20x - 12\)[/tex] are [tex]\([2, 13, 20, -12]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(2, 13, 20, -12\)[/tex].
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply by [tex]\(-4\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(2 \cdot (-4) = -8\)[/tex].
- Write the result under the next coefficient and add: [tex]\(13 + (-8) = 5\)[/tex].
- Multiply by [tex]\(-4\)[/tex] again: [tex]\(5 \cdot (-4) = -20\)[/tex].
- Write the result under the next coefficient and add: [tex]\(20 + (-20) = 0\)[/tex].
- Multiply by [tex]\(-4\)[/tex] again: [tex]\(0 \cdot (-4) = 0\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-12 + 0 = -12\)[/tex].
The quotient is [tex]\(2x^2 + 5x\)[/tex] and the remainder is [tex]\(-12\)[/tex]. So:
[tex]\[ \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} = (2x^2 + 5x) + \frac{-12}{x + 4} \][/tex]
In summary:
[tex]\[ (a) \frac{x^2 - 17x - 33}{x + 2} = (x - 19) + \frac{5}{x + 2} \][/tex]
[tex]\[ (b) \frac{2x^2 - 9x + 19}{x - 6} = (2x + 3) + \frac{37}{x - 6} \][/tex]
[tex]\[ (c) \frac{x^3 - 2x^2 + 7x - 25}{x - 3} = (x^2 + x + 10) + \frac{5}{x - 3} \][/tex]
[tex]\[ (d) \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} = (2x^2 + 5x) + \frac{-12}{x + 4} \][/tex]
### Part (a)
[tex]\[ \frac{x^2 - 17x - 33}{x + 2} \][/tex]
Here, we are dividing [tex]\(x^2 - 17x - 33\)[/tex] by [tex]\(x + 2\)[/tex].
1. Set [tex]\(x + 2 = 0\)[/tex] which gives [tex]\(x = -2\)[/tex], the value we will use for synthetic division.
2. Coefficients of the polynomial [tex]\(x^2 - 17x - 33\)[/tex] are [tex]\( [1, -17, -33] \)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(1, -17, -33\)[/tex].
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply by [tex]\(-2\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(1 \cdot (-2) = -2\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-17 + (-2) = -19\)[/tex].
- Multiply by [tex]\(-2\)[/tex] again: [tex]\(-19 \cdot (-2) = 38\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-33 + 38 = 5\)[/tex].
The quotient is [tex]\(x - 19\)[/tex] and the remainder is [tex]\(5\)[/tex]. So:
[tex]\[ \frac{x^2 - 17x - 33}{x + 2} = (x - 19) + \frac{5}{x + 2} \][/tex]
### Part (b)
[tex]\[ \frac{2x^2 - 9x + 19}{x - 6} \][/tex]
Here, we are dividing [tex]\(2x^2 - 9x + 19\)[/tex] by [tex]\(x - 6\)[/tex].
1. Set [tex]\(x - 6 = 0\)[/tex] which gives [tex]\(x = 6\)[/tex].
2. Coefficients of the polynomial [tex]\(2x^2 - 9x + 19\)[/tex] are [tex]\([2, -9, 19]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(2, -9, 19\)[/tex].
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply by [tex]\(6\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(2 \cdot 6 = 12\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-9 + 12 = 3\)[/tex].
- Multiply by [tex]\(6\)[/tex] again: [tex]\(3 \cdot 6 = 18\)[/tex].
- Write the result under the next coefficient and add: [tex]\(19 + 18 = 37\)[/tex].
The quotient is [tex]\(2x + 3\)[/tex] and the remainder is [tex]\(37\)[/tex]. So:
[tex]\[ \frac{2x^2 - 9x + 19}{x - 6} = (2x + 3) + \frac{37}{x - 6} \][/tex]
### Part (c)
[tex]\[ \frac{x^3 - 2x^2 + 7x - 25}{x - 3} \][/tex]
Here, we are dividing [tex]\(x^3 - 2x^2 + 7x - 25\)[/tex] by [tex]\(x - 3\)[/tex].
1. Set [tex]\(x - 3 = 0\)[/tex] which gives [tex]\(x = 3\)[/tex].
2. Coefficients of the polynomial [tex]\(x^3 - 2x^2 + 7x - 25\)[/tex] are [tex]\([1, -2, 7, -25]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(1, -2, 7, -25\)[/tex].
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply by [tex]\(3\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(1 \cdot 3 = 3\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-2 + 3 = 1\)[/tex].
- Multiply by [tex]\(3\)[/tex] again: [tex]\(1 \cdot 3 = 3\)[/tex].
- Write the result under the next coefficient and add: [tex]\(7 + 3 = 10\)[/tex].
- Multiply by [tex]\(3\)[/tex] again: [tex]\(10 \cdot 3 = 30\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-25 + 30 = 5\)[/tex].
The quotient is [tex]\(x^2 + x + 10\)[/tex] and the remainder is [tex]\(5\)[/tex]. So:
[tex]\[ \frac{x^3 - 2x^2 + 7x - 25}{x - 3} = (x^2 + x + 10) + \frac{5}{x - 3} \][/tex]
### Part (d)
[tex]\[ \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} \][/tex]
Here, we are dividing [tex]\(2x^3 + 13x^2 + 20x - 12\)[/tex] by [tex]\(x + 4\)[/tex].
1. Set [tex]\(x + 4 = 0\)[/tex] which gives [tex]\(x = -4\)[/tex].
2. Coefficients of the polynomial [tex]\(2x^3 + 13x^2 + 20x - 12\)[/tex] are [tex]\([2, 13, 20, -12]\)[/tex].
Using synthetic division:
- Write down the coefficients: [tex]\(2, 13, 20, -12\)[/tex].
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply by [tex]\(-4\)[/tex] (value of [tex]\(x\)[/tex]): [tex]\(2 \cdot (-4) = -8\)[/tex].
- Write the result under the next coefficient and add: [tex]\(13 + (-8) = 5\)[/tex].
- Multiply by [tex]\(-4\)[/tex] again: [tex]\(5 \cdot (-4) = -20\)[/tex].
- Write the result under the next coefficient and add: [tex]\(20 + (-20) = 0\)[/tex].
- Multiply by [tex]\(-4\)[/tex] again: [tex]\(0 \cdot (-4) = 0\)[/tex].
- Write the result under the next coefficient and add: [tex]\(-12 + 0 = -12\)[/tex].
The quotient is [tex]\(2x^2 + 5x\)[/tex] and the remainder is [tex]\(-12\)[/tex]. So:
[tex]\[ \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} = (2x^2 + 5x) + \frac{-12}{x + 4} \][/tex]
In summary:
[tex]\[ (a) \frac{x^2 - 17x - 33}{x + 2} = (x - 19) + \frac{5}{x + 2} \][/tex]
[tex]\[ (b) \frac{2x^2 - 9x + 19}{x - 6} = (2x + 3) + \frac{37}{x - 6} \][/tex]
[tex]\[ (c) \frac{x^3 - 2x^2 + 7x - 25}{x - 3} = (x^2 + x + 10) + \frac{5}{x - 3} \][/tex]
[tex]\[ (d) \frac{2x^3 + 13x^2 + 20x - 12}{x + 4} = (2x^2 + 5x) + \frac{-12}{x + 4} \][/tex]
