Divide
(42b^3 + 23b^2 + 38b + 49)/ (6b + 5)
See my reply to your re-post: unless I've slipped up myself I think you may have made a typing error with the constant term of the numerator, since 40 would make the numerator exactly divisible by (6b+5), but 49 doesn't.
To divide the polynomial by the binomial, we can use long division. Here are the steps:
1. Divide the first term of the polynomial (42b^3) by the first term of the binomial (6b). Write the result (7b^2) on top.
7b^2
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
- (42b^3 + 35b^2)
2. Multiply the result from step 1 (7b^2) by the binomial (6b + 5) and write the result (42b^2 + 35b^2) under the polynomial. Align like terms.
7b^2
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
-(42b^3 + 35b^2)
23b^2 + 38b + 49
3. Subtract the polynomial (42b^2 + 35b^2) from the polynomial (23b^2 + 38b + 49) and write the result ( -12b^2 + 38b + 49) below the line.
7b^2
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
-(42b^3 + 35b^2)
______________
-12b^2 + 38b + 49
4. Repeat the process for the new polynomial (-12b^2 + 38b + 49). Divide the first term of the new polynomial (-12b^2) by the first term of the binomial (6b). Write the result (-2b) on top.
7b^2 - 2b
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
-(42b^3 + 35b^2)
______________
-12b^2 + 38b + 49
-(-12b^2 - 10b)
5. Multiply the result from step 4 (-2b) by the binomial (6b + 5) and write the result (-12b^2 - 10b) under the polynomial (-12b^2 + 38b + 49). Align like terms.
7b^2 - 2b
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
-(42b^3 + 35b^2)
______________
-12b^2 + 38b + 49
-(-12b^2 - 10b)
______________
48b + 49
6. Subtract the polynomial (-12b^2 - 10b) from the polynomial (-12b^2 + 38b + 49) and write the result (48b + 49) below the line.
7b^2 - 2b
______________
6b + 5 | 42b^3 + 23b^2 + 38b + 49
-(42b^3 + 35b^2)
______________
-12b^2 + 38b + 49
-(-12b^2 - 10b)
______________
48b + 49
7. At this point, we have no more terms to divide. The remainder is (48b + 49).
Therefore, the quotient is 7b^2 - 2b and the remainder is 48b + 49.
To divide the polynomial (42b^3 + 23b^2 + 38b + 49) by the polynomial (6b + 5), we can use the long division method. Here's how you can do it step by step:
Step 1: Arrange the polynomials in descending order of powers of b.
Dividend: 42b^3 + 23b^2 + 38b + 49.
Divisor: 6b + 5.
Step 2: Divide the first term of the dividend (42b^3) by the first term of the divisor (6b).
42b^3 / 6b = 7b^2.
Step 3: Multiply the divisor by the quotient obtained in the previous step (7b^2).
7b^2 * (6b + 5) = 42b^3 + 35b^2.
Step 4: Subtract the result obtained in step 3 from the dividend.
(42b^3 + 23b^2 + 38b + 49) - (42b^3 + 35b^2).
Removing the parentheses, we get:
42b^3 + 23b^2 + 38b + 49 - 42b^3 - 35b^2.
Combine like terms:
(42b^3 - 42b^3) + (23b^2 - 35b^2) + 38b + 49.
Simplifying further:
-12b^2 + 38b + 49.
Step 5: Divide the first term of the simplified dividend (-12b^2) by the first term of the divisor (6b).
-12b^2 / 6b = -2b.
Step 6: Multiply the divisor by the quotient obtained in the previous step (-2b).
-2b * (6b + 5) = -12b^2 - 10b.
Step 7: Subtract the result obtained in step 6 from the simplified dividend.
(-12b^2 + 38b + 49) - (-12b^2 - 10b).
Removing the parentheses, we get:
-12b^2 + 38b + 49 + 12b^2 + 10b.
Combine like terms:
(-12b^2 + 12b^2) + (38b + 10b) + 49.
Simplifying further:
48b + 49.
Step 8: Divide the first term of the simplified dividend (48b) by the first term of the divisor (6b).
48b / 6b = 8.
Step 9: Multiply the divisor by the quotient obtained in the previous step (8).
8 * (6b + 5) = 48b + 40.
Step 10: Subtract the result obtained in step 9 from the simplified dividend.
(48b + 49) - (48b + 40).
Removing the parentheses, we get:
48b + 49 - 48b - 40.
Combine like terms:
(48b - 48b) + (49 - 40).
Simplifying further:
9.
The remainder is 9.
Therefore, the division of (42b^3 + 23b^2 + 38b + 49) by (6b + 5) is equal to 7b^2 - 2b + 8 with a remainder of 9.