1) To find the polynomial, we will perform polynomial long division.
Step 1: Write the dividend (the polynomial being divided) and divisor:
Dividend: Let's call it P(x)
Divisor: (x - 3)
Step 2: Divide the first term of the dividend (in this case, x^2) by the first term of the divisor (which is x), which gives us x.
Write this as the first term in the quotient.
Step 3: Multiply the divisor by x (the term we just obtained) and subtract it from the dividend:
(x - 3) * x = x^2 - 3x
P(x) - (x^2 - 3x) = 2x - 5
Step 4: Bring down the next term from the dividend, which is 2x:
Dividend: 2x - 5
Step 5: Repeat steps 2 to 4 until there are no more terms in the dividend.
Divide (2x - 5) by (x - 3):
Step 2: (2x) / (x) = 2
Write 2 as the next term of the quotient.
Step 3: Multiply (x - 3) by 2 and subtract it from (2x - 5):
(2)(x - 3) = 2x - 6
(2x - 5) - (2x - 6) = 1
Step 4: There are no more terms to bring down.
The quotient is x^2 + 2x - 5, and the remainder is 1. Therefore, the original polynomial is:
(x - 3)(x^2 + 2x - 5) + 1.
2) Let's go through each division one by one:
a) (2x^2 + 29x - x^3 - 40) / (-3 + x)
To find the quotient and remainder, divide (2x^2 + 29x - x^3 - 40) by (-3 + x).
b) (6 + 7x - 11x^2 - 2x^3) / (x + 9)
To find the quotient and remainder, divide (6 + 7x - 11x^2 - 2x^3) by (x + 9).
c) (x^3 - 2x^2 + 4x + 150) / (x^2 + 2x - 3)
To find the quotient and remainder, divide (x^3 - 2x^2 + 4x + 150) by (x^2 + 2x - 3).
d) (3x^3 + 2x^2 - 11x - 12) / (x + 1)
To find the quotient and remainder, divide (3x^3 + 2x^2 - 11x - 12) by (x + 1).
e) (x^2 + x^2y - 9xy^2 - 9y^3) / (x + y)
To find the quotient and remainder, divide (x^2 + x^2y - 9xy^2 - 9y^3) by (x + y).