Who invented factor theorem

Divide 15 by 6. What answer do you get? By using the simple division process, we find that the quotient is 2 and the remainder is 3. On the other hand, when we divide 12 by 6, we get a quotient of 2 and remainder 0. In this case, we say that 6 is a factor of 12 OR 12 is a multiple of 6. Here, 1 is not divisible by x. So, we stop the division here and note that 1 is the remainder.

So, the result of the division is,. We arrange the terms on the descending order of their degrees. This is the first term of the quotient. Repeat step 2 to get the next term of the quotient. Remember: This process continues until the degree of the new dividend is less than the degree of the divisor. Hence, we have quotient: 3x — 2 and remainder: 1. Here we say that p x divided by g x , gives q x as quotient and r x as remainder. In other words, the remainder obtained on dividing a polynomial by another is the same as the value of the dividend polynomial at the zero of the divisor polynomial.

This brings us to the first theorem of this article. If p x is divided by the linear polynomial x — a , then the remainder is p a. Proof: p x is a polynomial with a degree greater than or equal to one. So, we can write,. Now, the degree of x — a is 1. Also, since r x is the remainder, its degree is less than the degree of the divisor: x — a. We set up for synthetic division.

A couple of things about the last example are worth mentioning. First, the extension of the synthetic division tableau for repeated divisions will be a common site in the sections to come. Typically, we will start with a higher order polynomial and peel off one zero at a time until we are left with a quadratic, whose roots can always be found using the Quadratic Formula.

We can certainly put the Factor Theorem to the test and continue the synthetic division tableau from above to see what happens. It may surprise and delight the reader that, in theory, all polynomials can be reduced to this kind of factorization.

We leave that discussion to Section 3. Our final theorem in the section gives us an upper bound on the number of real zeros. The next section provides us some tools which not only help us determine where the real zeros are to be found, but which real numbers they may be.

We close this section with a summary of several concepts previously presented. You should take the time to look back through the text to see where each concept was first introduced and where each connection to the other concepts was made. The following statements are equivalent:. Carl Stitz , Ph. To see how it works in the case of polynomials, let us consider the following example with two polynomials:.

On dividing polynomials , the quotient polynomial and the remainder are:. We have. Once again, this has turned out to be equal to the remainder we calculated using the division algorithm.

Example 1: Casey is solving a polynomial expression. We will use the remainder theorem: we will substitute the zero of q x into the polynomial p x to find the remainder r:. What can you infer from your answer? This means that if p x is divided by the linear polynomial q x : x - 2 there will be no remainder! Thus, we can conclude that q x is a factor of p x. In general, whenever a polynomial is divided by a linear divisor and the remainder is 0, the linear divisor must be a factor of the polynomial.

Find the remainder polynomial and the quotient when a x is divided by b x using the remainder theorem. Let us try to find the same using the remainder theorem.

Up to notational conventions polynomial division algorithm was first described by an Arabic mathematician al-Samawal , who can also be credited with defining what we call polynomials today, see Who invented short and long division? In light of that the factorization theorem does not predate polynomial division.

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