We want to divide two polynomials.
(x^3 - 2x^2 + 25x - 50)÷ (x-2)
We can use a generic rectangle. Let's set it up!
Empty blue squares in the top row represent the quotient. The first column is the divisor. Let's perform the division step by step. We will fill the first white square with the monomial of the highest degree from the dividend.
We want to find the value that should be in the square above x^3. Let's label it ?. This is the polynomial that gives x^3 when multiplied by x.
x * ? = x^3 ⇕ ? = x^2
We will fill in this square with x^2. In the square below x^3, we write the value that we just found multiplied by the second term of the divisor. In our exercise, this would be is -2 * x^2 = -2x^2.
The circled terms must add up to the next monomial in the dividend, -2x^2. Our circle already adds up to -2x^2, so we fill it in with a 0.
We continue by following the same steps with the new terms. Let's see the whole process!
The last square is the remainder. In our case, the remainder is 0, which means that x^3 - 2x^2 + 25x -50 divided by x-2 equals x^2 + 25 with no remainder.
Alternative Solution
Factoring the Polynomial
Another way to find the quotient of two polynomials is to try to factor out the divisor. Let's try it!
We found that x^3 - 2x^2 + 25x -50 is equivalent to (x-2)(x^2 + 25). This means that the quotient of x^3 - 2x^2 + 25x -50 and x-2 is equal to x^2 + 25. x^3 - 2x^2 + 25x -50 = (x-2)(x^2 + 25) ⇕
(x^3 - 2x^2 + 25x -50) ÷ (x-2) = x^2 + 25
