Find Quotient and the remainder
X^3+216a^3 divides by x+6a
You could do a long algebraic division, make sure you have
(x^3 + 0x^2 + 0x + 216a^3) ÷ (x + 6a)
or
you could recognize the sum of cubes factoring
x^3 + 216a^3
= x^3 + (6a)^3
= (x+6a)(x^2 - 6ax + 36a^2) , with 0 remainder
expand it to see that I am right
********** x^2 - 6 a x + 36 a^2
*********--------------------------------------
x + 6 a | x^3 + 0 x^2 + 0 x + 216 a^3
********* x^3 + 6 a x^2
********------------------
****************-6 a x^2 + 0 x + 216 a^3
****************-6 a x^2 -36 a^2 x
**************-----------------------------------
*************************36 a^2 x + 216 a^3
*************************36 a^2 x + 216 a^3
************************* -----------------------
*********************************** 0
To find the quotient and the remainder when dividing a polynomial by another polynomial, we can use polynomial long division. Here's how you can find the quotient and remainder for the given division.
Step 1: Arrange the Dividend and Divisor
Write the dividend (x^3 + 216a^3) and divisor (x + 6a) in descending order of powers:
Dividend: x^3 + 216a^3
Divisor: x + 6a
Step 2: Divide the First Term
Divide the highest power term of the dividend (x^3) by the highest power term of the divisor (x). The result will be the first term of the quotient:
Quotient: x^2 (since x^3 / x = x^2)
Step 3: Multiply and Subtract
Multiply the divisor (x + 6a) by the quotient term (x^2), and subtract that from the dividend:
(x^2)(x + 6a) = x^3 + 6ax^2
Dividend - (x^3 + 6ax^2) = (216a^3) - (6ax^2)
Step 4: Bring Down the Next Term
Bring down the next term from the dividend, which is -6ax^2:
New Dividend: (216a^3) - (6ax^2)
Step 5: Divide the First Term Again
Divide the highest power term of the new dividend (-6ax^2) by the highest power term of the divisor (x). The result will be the next term of the quotient:
Quotient: x^2 - 6a (since -6ax^2 / x = -6a)
Step 6: Multiply and Subtract Again
Multiply the divisor (x + 6a) by the new quotient term (x^2 - 6a), and subtract that from the new dividend:
(x^2 - 6a)(x + 6a) = x^3 + 6ax^2 - 6ax - 36a^2
New Dividend - (x^3 + 6ax^2 - 6ax - 36a^2) = (216a^3) - (- 6ax - 36a^2)
Step 7: Bring Down the Last Term
Bring down the last term from the new dividend, which is (- 6ax - 36a^2):
New Dividend: (216a^3) - (- 6ax - 36a^2)
Step 8: Divide the First Term Once More
Divide the highest power term of the new dividend (- 6ax - 36a^2) by the highest power term of the divisor (x). The result will be the last term of the quotient:
Quotient: x^2 - 6a + 6a^2 (since - 6ax / x = - 6a)
Step 9: Multiply and Subtract for the Final Time
Multiply the divisor (x + 6a) by the last quotient term (x^2 - 6a + 6a^2), and subtract that from the new dividend:
(x^2 - 6a + 6a^2)(x + 6a) = x^3 + 6ax^2 - 6ax - 36a^2 + 6ax + 36a^2 + 36a^3
New Dividend - (x^3 + 6ax^2 - 6ax - 36a^2 + 6ax + 36a^2 + 36a^3) = 0
Step 10: Determine the Quotient and Remainder
Since the subtraction resulted in zero, there is no remainder. Therefore, the quotient is:
Quotient: x^2 - 6a + 6a^2
In summary, the quotient when dividing x^3 + 216a^3 by x + 6a is x^2 - 6a + 6a^2, and there is no remainder.