How do you simplify #(2x^(3)-4x^(2)+12x+18)/(x+1)#?

To simplify the expression (2x^(3)-4x^(2)+12x+18)/(x+1), you can use polynomial long division or synthetic division. However, I will demonstrate the process using polynomial long division.

Step 1: Divide the first term of the numerator (2x^(3)) by the first term of the denominator (x). The result is 2x^(2).

Step 2: Multiply the entire denominator (x+1) by the quotient obtained in step 1 (2x^(2)). The result is 2x^(2)(x+1) = 2x^(3) + 2x^(2).

Step 3: Subtract the result obtained in step 2 from the numerator (2x^(3)-4x^(2)+12x+18). This gives us (-4x^(2)+12x+18) - (2x^(3) + 2x^(2)) = -2x^(3) - 6x^(2) + 12x + 18.

Step 4: Repeat steps 1-3 with the new expression (-2x^(3) - 6x^(2) + 12x + 18) as the numerator.

Step 5: Divide the first term of the new numerator (-2x^(3)) by the first term of the denominator (x). The result is -2x^(2).

Step 6: Multiply the entire denominator (x+1) by the quotient obtained in step 5 (-2x^(2)). The result is -2x^(2)(x+1) = -2x^(3) - 2x^(2).

Step 7: Subtract the result obtained in step 6 from the new numerator (-2x^(3) - 6x^(2) + 12x + 18). This gives us (-6x^(2) + 12x + 18) - (-2x^(3) - 2x^(2)) = -4x^(2) + 12x + 18.

Step 8: Repeat steps 1-7 with the new expression (-4x^(2) + 12x + 18) as the numerator.

Step 9: Divide the first term of the new numerator (-4x^(2)) by the first term of the denominator (x). The result is -4x.

Step 10: Multiply the entire denominator (x+1) by the quotient obtained in step 9 (-4x). The result is -4x(x+1) = -4x^(2) - 4x.

Step 11: Subtract the result obtained in step 10 from the new numerator (-4x^(2) + 12x + 18). This gives us (12x + 18) - (-4x^(2) - 4x) = 12x + 18 + 4x^(2) + 4x = 4x^(2) + 16x + 18.

Step 12: Repeat steps 1-11 with the new expression (4x^(2) + 16x + 18) as the numerator.

Step 13: Divide the first term of the new numerator (4x^(2)) by the first term of the denominator (x). The result is 4x.

Step 14: Multiply the entire denominator (x+1) by the quotient obtained in step 13 (4x). The result is 4x(x+1) = 4x^(2) + 4x.

Step 15: Subtract the result obtained in step 14 from the new numerator (4x^(2) + 16x + 18). This gives us (16x + 18) - (4x^(2) + 4x) = 16x + 18 - 4x^(2) - 4x = -4x^(2) + 12x + 18.

Step 16: Repeat steps 1-15 with the new expression (-4x^(2) + 12x + 18) as the numerator.

Step 17: Divide the first term of the new numerator (-4x^(2)) by the first term of the denominator (x). The result is -4x.

Step 18: Multiply the entire denominator (x+1) by the quotient obtained in step 17 (-4x). The result is -4x(x+1) = -4x^(2) - 4x.

Step 19: Subtract the result obtained in step 18 from the new numerator (-4x^(2) + 12x + 18). This gives us (12x + 18) - (-4x^(2) - 4x) = 12x + 18 + 4x^(2) + 4x = 4x^(2) + 16x + 18.

At this point, the new numerator (4x^(2) + 16x + 18) is equal to the remainder. Since the degree of the remainder (2) is less than the degree of the denominator (1), we have reached the end of the division.

Therefore, the simplified form of (2x^(3)-4x^(2)+12x+18)/(x+1) is 2x^(2) - 4x + 4 + (4x^(2) + 16x + 18)/(x+1).