Divide 15x5 − 3x3 − 9x2 by −3x2
There are different methods to approach this problem, but one possible way is to use polynomial long division. Here are the steps:
- Write the division problem in the standard form, with the dividend (15x5 − 3x3 − 9x2) inside the long division symbol and the divisor (−3x2) outside:
```
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
```
- Divide the leading term of the dividend (15x5) by the leading term of the divisor (−3x2) to get −5x3, which is the quotient for the first step. Write it above the long division symbol, aligned with the third degree term:
```
-5x^3
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
```
- Multiply the divisor (−3x2) by the quotient (−5x3) to get 15x5, then subtract it from the first two terms of the dividend (15x5 − 3x3) to obtain 3x3. Write this result below the line, aligned with the second degree term:
```
-5x^3
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
-15x^5 + 5x^3
--------------
3x^3
```
- Repeat the process by dividing the leading term of the remaining dividend (3x3) by the leading term of the divisor (−3x2) to get −x, which is the quotient for the second step. Write it above the line, aligned with the first degree term:
```
-5x^3 - x
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
-15x^5 + 5x^3
--------------
3x^3 - 9x^2
```
- Multiply the divisor (−3x2) by the quotient (−x) to get 3x2, then subtract it from the last two terms of the dividend (−9x2) to obtain 6x2. Write this result below the line, aligned with the zero degree term:
```
-5x^3 - x
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
-15x^5 + 5x^3
--------------
3x^3 - 9x^2
3x^3 + x^2
-----------
-8x^2
```
- Check if the degree of the remainder (−8x2) is less than the degree of the divisor (−3x2). Since it is, the division is complete and the quotient is given by the terms above the line, in decreasing order of degree:
```
-5x^3 - x
___________________________
-3x2 | 15x^5 - 3x^3 - 9x^2
-15x^5 + 5x^3
--------------
3x^3 - 9x^2
3x^3 + x^2
-----------
-8x^2
Quotient: -5x^3 - x
Remainder: -8x^2
```
Therefore, the result of dividing 15x5 − 3x3 − 9x2 by −3x2 is:
```
-5x^3 - x - 8x^2 / (-3x^2)
or
-5x^3 / (-3x^2) - x / (-3x^2) - 8x^2 / (-3x^2)
or
(5/3)x^2 + (1/3)x + (8/3) (with positive powers of x)
```
So the quotient is (5/3)x^2 + (1/3)x + (8/3) , and the remainder is -8x^2.
To divide 15x^5 − 3x^3 − 9x^2 by -3x^2, we can use polynomial long division. Here are the step-by-step instructions:
Step 1: Write the dividend (15x^5 − 3x^3 − 9x^2) and the divisor (-3x^2) in the long division format:
___________________________________
-3x^2 | 15x^5 - 3x^3 - 9x^2
Step 2: Divide the first term of the dividend (15x^5) by the first term of the divisor (-3x^2).
The result is -5x^3. Write this on top of the long division symbol:
-5x^3
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
Step 3: Multiply the divisor (-3x^2) by the quotient (-5x^3), and write the result below the dividend, making sure to align like terms:
-5x^3 * -3x^2 = 15x^5
-5x^3
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
-15x^5
Step 4: Subtract the result obtained in step 3 from the dividend. Write the result below the line, bringing down the rest of the terms:
(15x^5 - 15x^5) = 0
-5x^3
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
-15x^5
___________
0 - 3x^3 - 9x^2
Step 5: Repeat steps 2 to 4 with the updated dividend (0 - 3x^3 - 9x^2) and the divisor (-3x^2):
Divide the first term of the new dividend (-3x^3) by the first term of the divisor (-3x^2).
The result is x. Write this on top of the long division symbol:
-5x^3 + x
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
-15x^5
___________
0 - 3x^3 - 9x^2
0 + 3x^3
Step 6: Multiply the divisor (-3x^2) by the quotient (x), and write the result below the new dividend, making sure to align like terms:
x * -3x^2 = -3x^3
-5x^3 + x
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
-15x^5
___________
0 - 3x^3 - 9x^2
0 + 3x^3
______________
0
Step 7: Subtract the result obtained in step 6 from the new dividend. Write the result below the line and bring down any remaining terms:
(0 - 3x^3) = -3x^3
-5x^3 + x
_______________
-3x^2 | 15x^5 - 3x^3 - 9x^2
-15x^5
___________
0 - 3x^3 - 9x^2
0 + 3x^3
______________
0 - 9x^2
0
Step 8: The new dividend (-9x^2) is not divisible by the divisor (-3x^2) anymore. Therefore, the long division is complete.
Therefore, the division result is -5x^3 + x with a remainder of -9x^2.