To divide the polynomials using long division, we start by dividing the highest degree term of the dividend (8a^3) by the highest degree term of the divisor (2a), which gives us 4a^2. We then multiply this result by the entire divisor (2a+1) and subtract this product from the dividend (8a^3+1).
```
4a^2
____________
2a+1 | 8a^3 + 0a^2 + 0a + 1
- (8a^3 + 4a^2)
________________
- 4a^2 + 0a + 1
```
Now, we bring down the next term from the dividend, which is 0a, giving us -4a^2+0a+1. Next, we divide the highest degree term of this new polynomial (-4a^2) by the highest degree term of the divisor (2a), which gives us -2a. We then multiply this result by the entire divisor (2a+1) and subtract this product from -4a^2+0a+1.
```
4a^2 - 2a
______________
2a+1 | 8a^3 + 0a^2 + 0a + 1
- (8a^3 + 4a^2)
_____________________
- 4a^2 + 0a + 1
+ 4a^2 - 2a
____________
-2a + 1
```
Now, we bring down the next term from the dividend, which is 0a, giving us -2a+1. Next, we divide the highest degree term of this new polynomial (-2a) by the highest degree term of the divisor (2a), which gives us -1. We then multiply this result by the entire divisor (2a+1) and subtract this product from -2a+1.
```
4a^2 - 2a - 1
__________________
2a+1 | 8a^3 + 0a^2 + 0a + 1
- (8a^3 + 4a^2)
_______________________
-4a^2 + 0a + 1
+ 4a^2 - 2a - 1
___________________
-2a + 0
```
Since we have a remainder of -2a+0, we can write the original polynomial as:
(8a^3+1) / (2a+1) = 4a^2 - 2a - 1 - (-2a+0) / (2a+1)
To check our answer by multiplication, we can multiply the quotient (4a^2 - 2a - 1) by the divisor (2a+1) and add the remainder (-2a+0). If the result is equal to the original dividend (8a^3+1), then our division is correct.
(4a^2 - 2a - 1) * (2a+1) + (-2a+0) = 8a^3 + 0a^2 + 0a + 1
Since this is equal to the original dividend, our division is correct.
Divide the polynomials by using long division. Check your answer by multiplication.
(8a^3+1)/(2a+1)
3 answers
What was that answer?
The quotient is 4a^2 - 2a - 1 and the remainder is -2a.
7 answers