Use the long division method to find the result when 3, x, cubed, plus, 5, x, squared, plus, 20, x, plus, 123x
3
+5x
2
+20x+12 is divided by 3, x, plus, 23x+2.
To solve this problem using long division, we need to divide 3x^3 + 5x^2 + 20x + 12 by 3x + 23x + 2.
First, we divide the highest degree term of the dividend by the highest degree term of the divisor:
3x^3 ÷ 3x = x^2
Next, we multiply the divisor by x^2 and subtract it from the dividend:
(x^2) * (3x + 23x + 2) = 3x^3 + 23x^3 + 2x^2
(3x^3 + 5x^2 + 20x + 12) - (3x^3 + 23x^3 + 2x^2) = -18x^3 + 3x^2 + 20x + 12
Now, we repeat the process by dividing the new highest degree term of the dividend (-18x^3) by the highest degree term of the divisor (3x):
-18x^3 ÷ 3x = -6x^2
Multiply the divisor by -6x^2 and subtract it from the remaining terms of the dividend:
(-6x^2) * (3x + 23x + 2) = -18x^3 - 138x^2 - 12x^2
(-18x^3 + 3x^2 + 20x + 12) - (-18x^3 - 138x^2 - 12x^2) = 141x^2 + 32x + 12
Now, we repeat the process again by dividing the new highest degree term of the dividend (141x^2) by the highest degree term of the divisor (3x):
141x^2 ÷ 3x = 47x
Multiply the divisor by 47x and subtract it from the remaining terms of the dividend:
(47x) * (3x + 23x + 2) = 141x^2 + 1081x + 94x
(141x^2 + 32x + 12) - (141x^2 + 1081x + 94x) = -1033x - 12
Now, we divide the remaining term (-1033x) by the highest degree term of the divisor (3x):
-1033x ÷ 3x = -344
Since there are no more terms left in the dividend, we have reached the end of the long division process.
Therefore, the result of the division is:
Quotient: x^2 - 6x^2 + 47x - 344
Remainder: -1033x - 12