How do you divide #(8r^3 + 27s^3) /( 4r^2 - 6rs + 9s^2)#?

Question

Sophie Adams · Accepted Answer

#(8r^3+27s^3)/(4r^2-6rs+9s^2)=(2r+3s)#

To divide #(8r^3+27s^3)/(4r^2-6rs+9s^2)#, we should first factorize numerator and denominator.

As numerator is of type #a^3+b^3#, its factots will be of type #(a+b)(a^2-ab-b^2)#

Hence #(8r^3+27s^3)=[(2r)^3+(3s)^3]#

= #(2r+3s){(2r)^2-(2r)*(3s)+(3s)^2}# or

= #(2r+3s){4r^2-6rs+9s^2}#

But the latter factor is just the denominator.

Hence, #(8r^3+27s^3)/(4r^2-6rs+9s^2)#

= #((2r+3s)cancel(4r^2-6rs+9s^2))/(cancel(4r^2-6rs+9s^2))=(2r+3s)#

Lily Anderson · Answer

undefined To divide (8r^3 + 27s^3) by (4r^2 - 6rs + 9s^2), you can use polynomial long division or synthetic division. Here, I'll demonstrate polynomial long division:

1. Divide the first term of the numerator by the first term of the denominator. This gives you the first term of the quotient.
2. Multiply the entire denominator by the first term of the quotient, and subtract this product from the numerator.
3. Bring down the next term from the numerator.
4. Repeat steps 1-3 until you have no more terms in the numerator or the degree of the remaining term in the numerator is less than the degree of the denominator.

Given:
Numerator: 8r^3 + 27s^3
Denominator: 4r^2 - 6rs + 9s^2

Step 1:
First term of the quotient: (8r^3) / (4r^2) = 2r

Step 2:
2r * (4r^2 - 6rs + 9s^2) = 8r^3 - 12r^2s + 18rs^2
Subtract this from the numerator:
(8r^3 + 27s^3) - (8r^3 - 12r^2s + 18rs^2) = 27s^3 + 12r^2s - 18rs^2

Step 3:
Bring down the next term from the numerator: 27s^3

Step 4:
Repeat steps 1-3:

Step 1:
(27s^3) / (4r^2) = 0 (because the degree of 27s^3 is less than the degree of 4r^2)

Step 2:
0 * (4r^2 - 6rs + 9s^2) = 0
Subtract this from the remaining term in the numerator:
27s^3 - 0 = 27s^3

Since there are no more terms in the numerator, the division is complete.

The quotient is 2r, and there is a remainder of 27s^3. Therefore, the division result is:

Quotient: 2r 
Remainder: 27s^3

Lincoln Anderson · Answer

undefined To divide $\frac{{8r^3 + 27s^3}}{{4r^2 - 6rs + 9s^2}}$, you can use polynomial long division or synthetic division. Here, I'll demonstrate polynomial long division:

1. Divide the first term of the numerator by the first term of the denominator. This gives you the first term of the quotient.
2. Multiply the entire denominator by this first term of the quotient.
3. Subtract the result from the numerator.
4. Bring down the next term from the numerator and repeat steps 1-3 until you've divided all terms.

Here's the process step by step:

1. Divide $8r^3$ by $4r^2$, which gives $2r$.
2. Multiply $2r$ by $4r^2 - 6rs + 9s^2$, which gives $8r^3 - 12rs + 18s^2$.
3. Subtract $8r^3 - 12rs + 18s^2$ from $8r^3 + 27s^3$, which leaves $12rs - 9s^2$.
4. Bring down the next term $27s^3$ and repeat.

Now, divide $12rs$ by $4r^2$, which gives $3s$.

Multiply $3s$ by $4r^2 - 6rs + 9s^2$, which gives $12rs - 18s^2 + 27s^3$.

Subtract $12rs - 18s^2 + 27s^3$ from $12rs - 9s^2$, which leaves $9s^2$.

Since the degree of $9s^2$ is less than the degree of $4r^2 - 6rs + 9s^2$, the division is complete.

Therefore, the quotient is $2r + 3s$ and the remainder is $9s^2$.