8x^3 + 27 divided by 4x^2 + 12x +9
To divide two polynomials, we can use polynomial long division. Here's how you can divide (8x^3 + 27) by (4x^2 + 12x + 9):
Step 1: Arrange the terms in descending order of powers
8x^3 + 0x^2 + 0x + 27 is divided by 4x^2 + 12x + 9
Step 2: Divide the first term of the numerator by the first term of the denominator
(8x^3 / 4x^2) = 2x
Step 3: Multiply the entire denominator by the quotient obtained in the previous step (2x)
(2x * (4x^2 + 12x + 9)) = 8x^3 + 24x^2 + 18x
Step 4: Subtract the previous result from the original numerator
(8x^3 + 27) - (8x^3 + 24x^2 + 18x) = -24x^2 - 18x + 27
Step 5: Repeat the process with the new result
Divide (-24x^2 - 18x + 27) by (4x^2 + 12x + 9)
(4x^2 / 4x^2) = 1
Multiply the entire denominator by the quotient obtained (1)
(1 * (4x^2 + 12x + 9)) = 4x^2 + 12x + 9
Subtract the previous result from the current numerator
(-24x^2 - 18x + 27) - (4x^2 + 12x + 9) = -28x^2 - 30x + 18
Step 6: Repeat the process with the new result
Divide (-28x^2 - 30x + 18) by (4x^2 + 12x + 9)
(-28x^2 / 4x^2) = -7
Multiply the entire denominator by the quotient obtained (-7)
(-7 * (4x^2 + 12x + 9)) = -28x^2 - 84x - 63
Subtract the previous result from the current numerator
(-28x^2 - 30x + 18) - (-28x^2 - 84x - 63) = 54x + 81
Step 7: We are left with a remainder (54x + 81) and a zero in the denominator (since it is a constant term). Since we cannot divide further, we have our final result:
The quotient is 2x + 1 with a remainder of (54x + 81).
Therefore, (8x^3 + 27) divided by (4x^2 + 12x + 9) can be expressed as:
2x + 1 + (54x + 81) / (4x^2 + 12x + 9)