Use long division to perform the division. (Express your answer as quotient + remainder/divisor.)
(x^4 + 7x^3 − 2x^2 + x − 1) / (x − 1)
visit calc101.com and click on the long division button to play around with polynomial division.
To perform long division to divide a polynomial by another polynomial, follow these steps:
Step 1: Arrange the polynomials in descending order according to their exponents.
The dividend is: x^4 + 7x^3 − 2x^2 + x − 1
The divisor is: x − 1
Step 2: Start dividing by the highest power of x. In this case, it's the term x^4.
Divide x^4 by x to get x^3. Write the result above x^3.
Step 3: Multiply the divisor (x − 1) with the result (x^3) and write the product below the dividend.
(x^3) * (x − 1) = x^4 − x^3
Step 4: Subtract the obtained product from the dividend.
(x^4 + 7x^3 − 2x^2 + x − 1) − (x^4 − x^3) = 7x^3 − 2x^2 + x − 1 + x^3
Simplifying, we have:
7x^3 − 2x^2 + x − 1 + x^3 = 8x^3 − 2x^2 + x − 1
Step 5: Bring down the next term. In this case, bring down the term -2x^2.
The polynomial becomes: 8x^3 − 2x^2 + x − 1
Step 6: Repeat steps 2-5 until you have brought down all terms and there are no more divisions to be made.
Divide x^3 by x to get x^2. Write the result above x^2.
Multiply the divisor (x − 1) with the result (x^2) and write the product below the current polynomial.
(x^2) * (x − 1) = x^3 − x^2
Subtract the obtained product from the current polynomial.
(8x^3 − 2x^2 + x − 1) − (x^3 − x^2) = 8x^3 − 2x^2 + x − 1 − (x^3 − x^2)
Simplifying, we have:
8x^3 − 2x^2 + x − 1 − (x^3 − x^2) = 7x^3 − x^2 + x − 1
Bring down the next term, which is x.
The polynomial becomes: 7x^3 − x^2 + x − 1
Divide 7x^3 by x to get 7x^2. Write the result above x.
Multiply the divisor (x − 1) with the result (7x^2) and write the product below the current polynomial.
(7x^2) * (x − 1) = 7x^3 − 7x^2
Subtract the obtained product from the current polynomial.
(7x^3 − x^2 + x − 1) − (7x^3 − 7x^2) = 7x^3 − x^2 + x − 1 − (7x^3 − 7x^2)
Simplifying, we have:
7x^3 − x^2 + x − 1 − (7x^3 − 7x^2) = 6x^2 + x − 1
Bring down the final term, which is -1.
The polynomial becomes: 6x^2 + x − 1 − 1
Divide 6x^2 by x to get 6x. Write the result above x.
Multiply the divisor (x − 1) with the result (6x) and write the product below the current polynomial.
(6x) * (x − 1) = 6x^2 − 6x
Subtract the obtained product from the current polynomial.
(6x^2 + x − 1) − (6x^2 − 6x) = 6x^2 + x − 1 − (6x^2 − 6x)
Simplifying, we have:
6x^2 + x − 1 − (6x^2 − 6x) = 7x + 1
Since there are no more terms to bring down, the division is complete.
The final result is: x^3 + x^2 + 7x + 6 + (7x + 1) / (x - 1)