WHAT IS THE FACTOR THEOREM?
In order to factor the polynomials and to find the n roots of the polynomials, we mainly use the factor theorem. If we want to analyze polynomial equations, then factor theorem can be very helpful. Factoring can useful in real life too, when we are dividing and quantity into equal pieces, comparing prices, exchanging money, or understanding time.
FACTOR THOREM
A special kind of polynomial remainder theorem which links the factors of a polynomial and its zeroes is known as the factor theorem. All the known zeroes are removed from a given polynomial equation by the factor theorem, just leaving all the unknown zeroes. As a result, the zeroes can be easily found, and the resultant polynomial has a lower degree.
Zero of a polynomial — It is essential for us to know about the zero or a root of the polynomial, before learning about the factor theorem. Only if g(a) = 0, we can say that y = a is a zero or root of a polynomial. In order to find the zeroes of the second- degree polynomial g(y) = y2 + 2y — 15, let us consider that example. We simply have to solve the equation by using the factorization quadratic equation method in order to do this. This can be done as –
y2 + 2y — 15 ⟹ (y + 5) (y — 5) ⟹ 0
⟹ y = -5 and y = 3. Thus, two zeroes or roots are had by this second degree polynomial y2 + 2y — 15, that are 3 and -5.
Factor theorem formula — (y — a) can be considered as a factor of the polynomial g(y) of degree n ≥ 1, as per the factor theorem, if and only if g(a) = 0. ‘a’ here, is any real number. g(y) = (y — a) q(y) is the formula of the factor theorem. The fact that all the following statement apply for any polynomial g(y), is important to remember — (y — a) is a factor of g(y); g(a) = 0; The remainder becomes zero when g(y) is divided by (y — a); The solution to g(y) = 0 is a and the zero of the function g(y) is a.
PROOF OF FACTOR THEOREM
We need to first consider a polynomial g(y) which is being divided by (y — a), only if g(a) = 0, if we want to prove the factor theorem. The given polynomial can be written as the produce of its quotient and its divisor, by using the division algorithm. Dividend = (divisor * quotient) + remainder. g(y) = (y — a) q(y) + remainder. Here, the dividend is g(y), the divisor is (y — a), and the quotient is q(y). The remainder is 0, if we substitute g(a). ⟹ g(y) = (y — a) q(y) + 0 ⟹ g(y) = (y — a) q(y)
Therefore, we can say that the factor of the polynomial g(y) is (y — a). The remainder theorem stats that a polynomial g(y) has a factor (y — a), only if g(a) = 0, that is, if a is a root, and in our situation, we can see that the factor theorem is the result of the remainder theorem.
HOW TO USE THE FACTOR THEOREM
In order to factor second- degree or quadratic polynomials, we usually use the factorization method. We use the procedure given below to factor the polynomial for higher degrees. The first step would be use the synthetic division of the polynomial method so as to divide the polynomial given g(y) by the binomial given (y — a). The second step would be to confirm is the remainder us zero, after completing the division. It’ll result in (y -a) not being a factor of g(y), if the remainder is not zero. The third step is to write the given polynomial as the product of (y — a) and the quadratic quotient q(y) by using the division algorithm. Next, you’ll have to factor the quotient further if it is possible. And then finally the given polynomial has to be expressed as the product of its factors.
DIFFERENCE BETWEEN FACTOR THEOREM AND REMAINDER THEOREM
Factor theorem and remainder theorem both refer to two different concepts, even though they are similar. The remainder of the division of a polynomial by a binomial with the value of a function at a point is related by the remainder theorem. The factors of a given polynomials to its zeroes is related to by the factor theorem.
If you want to learn more about factor theorem, remainder theorem, or any other such interesting and important topics in math’s and science, all you need to do is check out www.doubtnut.com . You will find all the material that you need for you to ace your studies. If you have any questions, or doubts, they will all be solved by toppers and experts in the subjects, on the site.
