Divide the polynomials by using long division. Check your answer by multiplication.
(8a^3+1)/(2a+1)
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.
What was that answer?
The quotient is 4a^2 - 2a - 1 and the remainder is -2a.
To divide the given polynomials (8a^3+1)/(2a+1) using long division, follow these steps:
Step 1: Ensure the polynomials are written in descending order of exponents. If not, rearrange them accordingly. The dividend is 8a^3 + 0a^2 + 0a + 1, and the divisor is 2a + 1.
Step 2: Divide the first term of the dividend by the first term of the divisor. In this case, 8a^3 ÷ 2a = 4a^2.
Step 3: Multiply the divisor (2a + 1) by the quotient obtained in the previous step (4a^2).
(4a^2) * (2a + 1) = 8a^3 + 4a^2.
Step 4: Subtract the product obtained in the previous step from the original dividend.
(8a^3 + 1) - (8a^3 + 4a^2) = -4a^2 + 1.
Step 5: Bring down the next term from the original dividend. The new dividend becomes -4a^2 + 1a + 1.
Step 6: Divide the first term of the new dividend (-4a^2) by the first term of the divisor (2a).
-4a^2 ÷ 2a = -2a.
Step 7: Multiply the divisor (2a + 1) by the quotient obtained in the previous step (-2a).
(-2a) * (2a + 1) = -4a^2 - 2a.
Step 8: Subtract the product obtained in the previous step from the new dividend.
(-4a^2 + 1a + 1) - (-4a^2 - 2a) = 3a + 1.
Step 9: Bring down the next term from the original dividend. The new dividend becomes 3a + 1.
Step 10: Divide the first term of the new dividend (3a) by the first term of the divisor (2a).
3a ÷ 2a = 1.5.
Step 11: Multiply the divisor (2a + 1) by the quotient obtained in the previous step (1.5).
(1.5) * (2a + 1) = 3a + 1.5.
Step 12: Subtract the product obtained in the previous step from the new dividend.
(3a + 1) - (3a + 1.5) = -0.5.
At this point, we are left with a remainder of -0.5.
To check the answer using multiplication, multiply the quotient (4a^2 - 2a + 1.5) by the divisor (2a + 1). If the result is equal to the dividend (8a^3 + 1), then the division is correct.
(4a^2 - 2a + 1.5) * (2a + 1) = 8a^3 + 1
Since the result is indeed equal to the dividend, the division is correct.