Try to divide 2x^3-7x^2+6x+1 by x-1. What are the steps that were repeated?
Divide, multiply, subtract, and bring down.
Practice makes perfect
Assume we want to divide a polynomial by a polynomial. To recall the steps of polynomial long division, we will try to divide 2x^3-7x^2+6x+1 by x-1 using this method.
Step 1
Before the actual division can be performed, we have to make sure that the dividend and divisor are written in standard form. If a term is not present in the dividend, we should add the missing term with a coefficient of 0.
Dividend: & 2x^3-7x^2+6x+1
Divisor: & x-1
The terms of each polynomial are in descending degree order, so they are already written in standard form. Additionally, the dividend has all terms present. We are ready to divide!
Step 2
We will divide the first term of the dividend by the first term of the divisor.
2x^3/x=2x^2
This is the first term of the quotient.
Step 3
Next we will multiply the first term of the quotient by the divisor.
Let's write our product, with a negative sign in front of it, under the dividend.
Step 4
Now we will subtract the product from the dividend.
Step 5
Finally, we will bring down the remaining terms of the dividend. The result of this operation is called the remainder.
How Should We Proceed?
To find the second term of the quotient, we will treat the remainder as the dividend and follow Steps 2 through 5. We divide the first term of the remainder by the first term of the divisor.
- 5x^2/x=- 5x
Now, we will multiply - 5x by the divisor.
We can subtract the product from the current dividend.
Then we bring down the remaining term.
The third term of the quotient can be found by following Steps 2 through 5 once more. This time the remainder x+1 will be treated as the dividend.
The remainder 2 is a constant, so its degree is 0. The degree of the divisor x-1 is 1. Since the degree of the remainder is less than the degree of the divisor, this is the end of the division.
Conclusion
Using our example, let's list the steps of polynomial long division.
Write the dividend and divisor in standard form. If a term is not present in the dividend, add the term with a coefficient of 0.
Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.
Multiply the first term of the quotient by the divisor and write the product under the dividend.
Subtract the product from the dividend.
Bring down the next terms of the dividend. The result of this operation is a remainder.
Repeat Steps 2 through 5 treating the remainder as the dividend. Stop when the degree of the remainder is less than the degree of the divisor.
We can see that the steps repeated when dividing two polynomials are Steps 2 through 5. These steps can be summarized as divide, multiply, subtract, and bring down.
Extra
Additional note
The step of bringing down terms could be omitted for polynomials. For numeric long division this makes sense, but it is a bit more complicated with polynomial long division. Yes, we bring down the terms but, because there are operations between each term, we can include this as part of the subtraction step.
