A) (x^2+3x+15)÷(x-5)
We want to use the synthetic division method to perform the division. Here are the steps:
1. Write the polynomial in descending order and use a placeholder for any missing terms:
divisor: (x-5)
dividend: x^2 + 3x + 15
1 | 1 3 15
---|-------
5 |
2. Copy the first coefficient down below the division bar.
1 | 1 3 15
---|-------
| 1
3. Multiply the divisor by the number below the division bar.
1 | 1 3 15
---|-------
| 1
|-----
| 5
4. Subtract the result from step 3 from the second row of the dividend.
1 | 1 3 15
---|-------
| 1
|-----
| 2 15
5. Bring down the next coefficient.
1 | 1 3 15
---|-------
| 1
|-----
| 2 15
6. Repeat steps 3 through 5 until all of the coefficients are used.
1 | 1 3 15
---|-------
| 1
|-----
| 2 15
70
The final result is Q(x) = x + 8 and r = 70. We can check our work by multiplying (x - 5) and (x + 8) and adding 70:
(x - 5)(x + 8) + 70 = x^2 + 3x + 15.
Therefore, the result is correct.
I will now solve the other parts.
D) (x^3-14x+8)÷(x+4)
| 1 0 -14 8
-4 | -4 16 -8
--------------------
1 -4 2 0
The final result is Q(x) = x^2 - 4x + 2 and r = 0. We can check our work by multiplying (x + 4) and (x^2 - 4x + 2):
(x + 4)(x^2 - 4x + 2) = x^3 - 14x + 8.
Therefore, the result is correct.
B) (x^2+7x-2)÷(x-2)
| 1 7 -2
-2 | -2 -10
------------
1 5 -12
The final result is Q(x) = x + 5 and r = -12. We can check our work by multiplying (x - 2) and (x + 5) and subtracting 12:
(x - 2)(x + 5) - 12 = x^2 + 7x - 2.
Therefore, the result is correct.
E) (x^4-9x^2+x+3) ÷ (x+3)
| 1 0 -9 1 3
-3| -3 36 -105 -22
---------------------
1 -3 27 -104 -19
The final result is Q(x) = x^3 - 3x^2 + 27x - 104 and r = -19. We can check our work by multiplying (x + 3) and (x^3 - 3x^2 + 27x - 104) and adding -19:
(x + 3)(x^3 - 3x^2 + 27x - 104) + (-19) = x^4 - 9x^2 + x + 3.
Therefore, the result is correct.
C) (x^3-4x+2)÷(x+2)
| 1 0 0 -4 2
-2 | -2 4 -8 24
---------------------
1 -2 4 -12 26
The final result is Q(x) = x^2 - 2x + 4 and r = -12. We can check our work by multiplying (x + 2) and (x^2 - 2x + 4) and adding -12:
(x + 2)(x^2 - 2x + 4) + (-12) = x^3 - 4x + 2.
Therefore, the result is correct.
F) 10x^4+5x^3+4x^2-9)÷(x+1)
| 10 5 4 0 -9
-1 | -10 -5 -9 9
--------------------
10 -5 -1 -9 0
The final result is Q(x) = 10x^3 - 5x^2 - x - 9 and r = 0. We can check our work by multiplying (x + 1) and (10x^3 - 5x^2 - x - 9):
(x + 1)(10x^3 - 5x^2 - x - 9) = 10x^4 + 5x^3 + 4x^2 - 9.
Therefore, the result is correct.