In this explainer, we will learn how to perform long division on polynomials.
Before we begin explaining the process of long division of polynomials, letβs first recall what it means to divide integers and how we apply long division to integers.
If we have two integers and , where is nonzero, then is the number that when multiplied by , gives . Hence, . We define the division of polynomials in the same way.
Definition: Division of Polynomials
We define the quotient of two polynomials by finding the polynomial whose product with the divisor yields the dividend.
For example, if we want to find , we can note that we want to find a polynomial that when multiplied by , gives . One way of finding this value is to expand the product . We have
Thus,
We can then ask, βHow do we find an expression for a quotient of polynomials in general?β
Letβs first consider an example involving integers. If we want to determine the value of , we first note that 5 does not go exactly into 17. We can instead employ long division to evaluate this quotient:
We first see that three 5s make 15 and that we cannot have any more 5s. Removing this from 17, we are left with a remainder of 2. We call 17 the dividend since it is being divided by the divisor 5, we call 3 the quotient, and we call 2 the remainder.
It is also worth noting that we say that divides if its remainder is 0 after division by . If we call the remainder after dividing by Β , then we can guarantee that . If this is not the case, then we can increase the value of the quotient.
The reason for this is that the process for long division rewrites the fraction. Letβs say that has a quotient and a remainder . Then goes into times and has a remainder . This means that . Hence,
In our numerical example, this reads
We are almost ready to apply this process to polynomials. Letβs say we want to divide a polynomial called the dividend by another polynomial called the divisor. We want to do this by subtracting multiples of the divisor from the dividend until we can no longer do so. To see how to do this, letβs consider an example.
Suppose we want to divide by . We need to determine how many multiples of we can remove from . Each multiple will be a polynomial since the products of polynomials are polynomials. We will call the polynomial that tells us the number of multiples of we remove the quotient. This means that we want to find a polynomial whose product with is as close to as possible.
Letβs write this division using the same long division notation we use for integers:
We note that the leading term of the dividend is and that the leading term of the divisor is . We then see that . We will write this in our quotient above the -term:
Remember that we want to subtract the product of the quotient and divisor from the dividend. We can do this term by term. If we multiply the divisor by , we get
Subtracting this from the dividend yields
We write this in the long division notation as follows:
Before we continue with the division, it may be useful to consider what we have actually shown. We have shown that . We can then divide through by , which gives
Thus, we have reduced the degree of the dividend. We can apply this process again, this time with as the dividend. The leading term of this new dividend is , and the leading term of the divisor is . So, , and we add this to the quotient as follows:
We want to remove the divisor from the dividend 3 times. This gives us
We can represent this using long division notation as follows:
In this explainer, we will only deal with cases where the remainder is the zero polynomial. However, it is worth noting that each time we perform this process, we are lowering the degree of the dividend by the degree of the divisor. We keep this process going until the remainder has a lower degree than the divisor. At this point when we divide their leading terms, we would get either zero or a variable raised to a negative exponent, which is not a monomial.
Using polynomial long division, we have shown that . So,
We can verify this by multiplying the quotient by the divisor . We have
This works as a nice check to make sure that the answer is correct.
We can summarize the working in the above example as follows:
Letβs now see an example of how to apply this process to divide two polynomials.
Example 1: Dividing Polynomials Using Polynomial Long Division
Find the quotient of divided by .
Answer
We can find an expression for using long division. The first step is to find the quotient of the leading terms of the dividend and the divisor. We note that . So, we write in the quotient, and we subtract from the dividend. We have that , so we see that
We calculate that
We can then add this onto our division:
We want to apply this process again, this time with as the dividend. We need to see how many times the divisor goes into the dividend . We do this by first dividing their leading terms. We see that . So, we add 3 to the quotient, and we then need to subtract from the dividend. We find that . Subtracting this gives us the following:
We find that .
Hence,
We can verify this answer by calculating .
In our next example, we will find the quotient of two polynomials where both of the polynomials are nonmonic.
Example 2: Dividing a Polynomial by a First-Degree Divisor to Find the Quotient
Find the quotient when is divided by .
Answer
We can divide two polynomials using polynomial long division. First, we need to divide their leading terms. We have . We add this into our quotient, and this means we need to subtract from our dividend. Since , we can do this as follows:
We calculate that . We have shown that
We can apply this process again to divide by . We divide the leading terms to get . Thus, we add to the quotient and subtract from the dividend as follows:
We then calculate that .
We need to apply this process one final time to fully divide the expression.
This time, the division of the leading terms gives us . We add this to the quotient. We then need to subtract from this new dividend. Since this is equal to the dividend, we get a value of 0:
Hence,
In our next example, we will divide a nonmonic cubic polynomial by a linear polynomial.
Example 3: Dividing a Cubic Polynomial by a First-Degree Divisor to Find the Quotient
Find the quotient when is divided by .
Answer
We can divide two polynomials using polynomial long division. First, we need to divide their leading terms. We have . We add this to our quotient, and this means we need to subtract from our dividend. Since , we can do this as follows:
We calculate that
Since the degree of is higher than or equal to the degree of the divisor, we now need to apply this process again, with as our new dividend.
We divide their leading terms to get , which we add to the quotient. We then need to subtract from the dividend. We can do this as follows:
We calculate that
Since the degree of this polynomial is equal to that of the divisor, we need to apply this process one final time, with as the dividend.
We divide their leading terms to get , which we then add to the quotient. We then need to subtract from the dividend. Since this is equal to the dividend, we will end up with a remainder of 0, as shown:
Hence, we have shown that the quotient that results from dividing by is . We could check this answer by calculating to ensure that we get our original cubic dividend.
In our next example, we will see how to apply polynomial long division to find a missing dimension in a given diagram.
Example 4: Determining a Dimension Using Polynomial Long Division
Given that the area of the rectangle in the diagram is , find an expression for the width of the rectangle.
Answer
We first recall that the area of rectangle is given by the product of its length and width. This means we can determine the width of this rectangle by dividing its area by its length. This means we need to divide a cubic polynomial by a linear polynomial. We can do this using polynomial long division.
We first divide their leading terms to get . We add this to the quotient and then subtract from the dividend. We note that and that , as shown:
Since the degree of is higher than or equal to the degree of the divisor, we need to apply this process again, with as the new dividend.
We first divide their leading terms to get , which we then add to the quotient. We now need to subtract from the divided. This gives us the following:
We calculate that
Since the degree of this polynomial is equal to the degree of the divisor, we need to apply this process one final time, with as the dividend.
We divide their leading terms to get , which we add to the quotient. We then note that . This is equal to the dividend, so when we subtract, we will get 0. This gives us the following:
Hence, , and the width of the rectangle is given by .
In our final example, we will use polynomial long division and the given divisibility of two polynomials to determine the value of an unknown coefficient.
Example 5: Finding the Value of a Constant That Makes a Polynomial Divisible
Find the value of that makes the expression divisible by .
Answer
We first recall that we say a polynomial is divisible by another polynomial if their division has a remainder of 0. This means we can apply polynomial long division to the given polynomials, and we know that the remainder must be the zero polynomial.
Before we apply polynomial long division, it is worth noting that the quintic polynomial is not given in descending powers of . We should always reorder the dividend to have descending powers of since we always want to remove the leading terms of the dividends. So, we will use as the dividend.
To apply polynomial long division, we first need to divide the leading terms. We get . We add this to the quotient and then subtract from the dividend. Since , we will also include the term to keep the columns with the same powers of . We have
We have calculated that
It is also worth noting that we add the term into our long division to keep the columns with the same powers of .
Since this is not zero, we need to apply this process again, with as the dividend.
We first divide the leading terms to get , and we add this to the quotient. Next, we need to subtract from the dividend. We note that . We can then subtract this from the dividend as follows:
We can calculate that
Since this is not zero, we need to apply this process again, with as the dividend.
We divide their leading terms to get , and we add this to our quotient. We then subtract from the dividend as follows:
We calculated that
Since we are told that the division is exact, we know that the remainder must be zero. Hence, .
We can solve this to see that .
Letβs finish by recapping some of the important points from this explainer.
Key Points
- We define the quotient of two polynomials by finding the polynomial whose product with the divisor yields the dividend.
- We can divide polynomials using long division.
- We can check our answer by multiplying the divisor by the quotient.
- If the remainder polynomial in the division of two polynomials is the zero polynomial, then we say that the divisor divides the dividend.
- We should always reorder the dividend to have descending powers of since we always want to remove the leading terms of the dividends.
