The polynomial long division aspects
A polynomial long division is similar to a long division of numbers. The polynomial long division method is used to write improper rational expressions as the sum of a proper rational expression and a polynomial. The polynomial long division can be done by hand because it separates a complex division into a smaller one. Polynomials are expressions which contain variables rose to any power and are multiplied by any type of number. When performing a polynomial long division you must know some vocabularies like the word “term” because they will be mentioned throughout the polynomial long division process. Some other vocabularies used in polynomial long division include variable and constant: the part of an algebraic equation that does not change. There is an also numeric and algebraic expression in polynomial long division. A numeric expression has operations and numbers and the algebraic expression is almost similar but it has variables. Another polynomial long division vocabulary is the quotient which is the result you get when you divide a number by another number. Other things that you need to note in polynomial long division calculations are:
- You should never write the polynomial long division mixed number the same way numerical mixed numbers are written. If you just indicate the fraction part of the polynomial, it will be interpreted as a multiplication.
- Be aware that various books format the polynomial long division differently. When you are writing the expressions on top of the division sign, some write the terms above the same degree term and others write it above the term being worked on. The difference is that the terms across the top are moved to the right but everything else is the same. You can choose the polynomial long division format that you prefer.
- For a polynomial long division to be successful, you have to write neatly and do not forget to change the signs when subtracting. You need to work carefully and line your columns well.
- The same way you carryout long division with plain numbers where you have one number that you divide into another number and then set up the division symbol and insert the two numbers and start calculating, is the same way polynomial long division works.
The polynomial long division algorithm implements the Euclidean division of polynomials, which begins from two polynomials A and B. A is the dividend (number being divided) and B is the divisor (the dividing number). If B is not a zero, you get the quotient Q and a remainder R such that A=BQ+R and either R is equal to zero or the degree of R is smaller than the degree of B. These are the conditions that uniquely define R and Q. This simply means that R and Q do not depend on the algorithm used to calculate them.
How to divide by a polynomial that has more than one term
The first step in the polynomial long division process is to ensure that the polynomial is expressed in a descending order. If there is a missing term, zero is used to fill it in. The second step in the polynomial long division is to divide the term with the highest power in the division symbol by a term which has the highest power outside the division symbol. The third step is to multiply the answer by the polynomial that is in front of the division symbol. The fourth step is subtracting and bringing down the next term and the fifth step is repeating steps 1,2,3 and 4 until you have no more terms to bring down. The sixth and final step in the polynomial long division process is writing the final answer. The term that remains after the last subtraction is the remainder and has to be written as a fraction in your final answer.
Example1: Divide: .
The solution to dividing x2 – 9x – 10 by x + 1 will be x-10. And since the remainder for this polynomial long division is zero the division is even. When you perform a regular division with numbers and find out that the division is even, this implies that the number you divided by is one of the factors of that number you are dividing. For example, if you divide 50 by 10, the answer will be a 5 without a remainder; this is because 10 is a factor of 50. In the case of the above polynomial long division, the zero remainder means that x + 1 is a factor of x2 – 9x – 10. You can confirm it by factoring the initial quadratic dividend, x2 – 9x – 10.
Applications of polynomials
A polynomial long division can be used when you have one or more roots of a polynomial. From the rational root theorem, if one root r of a given polynomial P(x) which has a degree n is provided, then you can use the polynomial long division method to factor P(x) in the form (x-r)(Q(x)) where Q(x) is a polynomial with a degree n-1. Q(x) is the quotient gotten from the polynomial long division process. And since r is the root of P(x) then the remainder must be zero.
Similarly, if you are given more than one root and a linear factor x-r is one of the roots and r can be divided to get Q(x), a linear term in another root s can be divided out of Q(x) and so on. Alternatively, all of them can be divided out at once. For instance, the linear factors x-s and x-r can be multiplied together to get a quadratic factor x2 − (r + s) x + rs which will then be divided into the initial polynomial P(x) to find the quotient of the degree n-2.
With this polynomial long division method, at times, all the roots of a polynomial of a degree that is more than four can be calculated though sometimes it is not possible. For instance, if the rational root theorem is used to find a single root of a quintic polynomial, then it can be used in obtaining a fourth-degree quotient and the explicit root formula of a fourth-degree polynomial can be used to get the remaining four roots.
The polynomial long division process can also be used to find the equation of tangent lines. The tangent to a graph of any function is defined by the polynomial P(x) at a certain point x=r. and if R(x) is the remainder after dividing P(x) by (x – r)2 then the tangent line equation at point x=r to the graph of a function y=P(x) is y=R(x) regardless of whether or not r is the polynomial root.
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