Dividing Polynomials Definitions and Examples
Introduction
In mathematics, division is the process of taking one number, called the dividend, and dividing it by another number, called the divisor. The answer to this division is called the quotient. In some cases, the quotient will be a whole number. However, more often than not, the quotient will be a fraction. When dividing polynomials, there are a few things to keep in mind. First and foremost, you can only divide if the divisor is a factor of the dividend. Secondly, you must use long division when dividing polynomials. Lastly, you should use synthetic division if the divisor is a linear factor of the dividend (meaning that it’s in the form of x-a). Don’t let long division intimidate you! It’s actually not as bad as it seems. In this blog post, we’ll go over all the steps necessary for dividing polynomials so that you can master this math skill in no time.
Dividing Polynomials
When it comes to polynomials, division is a bit different than the division you’re used to. In this lesson, we’ll cover what it means to divide polynomials and go over some examples.
Dividing polynomials is a process of breaking a polynomial up into smaller pieces. This is done by finding a factor of the polynomial and dividing it out. For example, if we have the polynomial x^2+5x+6, we can divide it by the factor (x+2) to get x+3.
Let’s look at another example. This time, we’ll divide the polynomial x^3-4x^2-7x+20 by the factor (x-1). We can do this by using long division or synthetic division. Long division is the more traditional method, but synthetic division is faster and easier once you get the hang of it.
What is Dividing Polynomials?
Dividing polynomials is the process of breaking a polynomial down into smaller pieces, or factors. This can be done using long division, synthetic division, or factoring. Long division is the most common method used to divide polynomials, and it is the method that will be described here.
To divide one polynomial by another, we first need to determine the order of the polynomials. The order of a polynomial is determined by the degree of the largest term. For example, in the polynomial 4x^3 + 3x^2 + 2x + 1, the degree of the largest term (4x^3) is 3, so the order of this polynomial is 3.
Once we know the order of both polynomials, we can begin long division. We start by dividing the leading term (the term with the highest degree) of the dividend (the number being divided) by the leading term of the divisor (the number doing the dividing). In our example, we would divide 4x^3 by x^3. This gives us a quotient of 4x^0 and a remainder of 3x^2 + 2x + 1.
Dividing Polynomials by Binomials
Dividing polynomials by binomials is a process by which we can determine the quotient of two polynomials when one polynomial is divided by another. This process is relatively simple, and can be done by following a few steps.
First, we need to determine the degree of the polynomials involved. The degree of a polynomial is the highest exponent of the variable in the equation. For example, in the equation x2+3x+5, the degree would be 2 since that is the highest exponent on the variable x.
Next, we need to identify the leading coefficients of each term. The leading coefficient is simply the numerical value in front of the variable with the highest exponent. In our example equation from before, 5 would be the leading coefficient since it is in front of x0.
Now that we have determined these values, we can begin to divide our polynomials. We will start by dividing the leading coefficients and then work our way down through each term, dividing as we go. In our example equation, we would first divide 5 by 3 to get 1 with a remainder of 2. We would then divide 1 (the new leading coefficient) into 2 (the old leading coefficient) to get 0 with a remainder of 2. Finally, we divide 0 into 2 (the new leading coefficient after division) to get 0 with a remainder of 2 again.
Dividing Polynomials Using Synthetic Division
When polynomials are divided, the goal is to find the quotient and the remainder. The quotient is the answer you would get if you were to divide the two polynomials using long division. The remainder is what is left over after division.
Synthetic division is a method of dividing polynomials that uses a simpler process than long division. To use synthetic division, you need to know the divisor and the dividend. The divisor is the number that you are dividing by, and the dividend is the number that you are dividing into.
To divide using synthetic division, write the divisor and dividend as follows:
Divisor: x – a
Dividend: xn + bxn-1 + cxd-2 + … + r
where n is the degree of the dividend, a is any number (except 0), b, c, … ,r are coefficients of respective terms in decreasing order of degree n, x ? a.
a | b | c | … | r
–a | x-a | x-a | … | x-a
——————————
b’| c’ | d’ | … | r’ <– Quotient terms in respectively decreasing order of degree n-1
Dividing Polynomials Examples
When dividing polynomials, we are looking for the quotient and the remainder. The quotient is the answer we are looking for when we divide, and the remainder is what is left over after division.
To divide polynomials, we use long division. This is similar to the long division you learned in elementary school, but with a few tweaks. First, we need to make sure that the divisor (the number being divided into) is a factor of the dividend (the number being divided). To do this, we use synthetic division.
Next, we divide using long division. We bring down the first term of the dividend, divide it by the divisor, and write the answer as the first term of the quotient. We then multiply the divisor by this answer and subtract it from the dividend. This gives us our next term in the quotient as well as our new dividend. We continue this process until we have no more terms in our dividend.
Here is an example:
We want to divide ?3+4?2+5?+6 by ?+2 .
factors of 6 are 1 , 2 , 3 , 6 so ?+2 is a factor of ?3+4?2+5?+6 .
Conclusion
Dividing polynomials is a relatively simple process, but one that can be confusing if you don’t understand the terminology. This article has provided you with a brief introduction to some of the key concepts involved in division of polynomials, as well as some worked examples to illustrate the process. If you still feel unsure about how to divide polynomials, be sure to seek out additional resources so that you can gain a better understanding.
