Factor Theorem

Factor theorem is a special kind of the polynomial remainder theorem that links the factor and zeros of a polynomial. The factor theorem removes all the known zeroes from a given polynomial equation and leaves all the unknown zeroes of the polynomial equation. The resultant polynomial has a lower degree in which the zeroes can be found easily.

Factor theorem is mainly used to factor the polynomial and to find the n roots of the polynomial. Factor theorem is very helpful for the analysis of polynomial equations. In real life, factoring can be useful, especially while exchanging money, dividing any quantity into equal pieces, understanding the time, and comparing prices.

ZERO OF THE POLYNOMIAL:

Before learning about the factor theorem, it is essential to know about the zero or in other words a root of the polynomial. We say that y = a is a root or zero of a polynomial g(y) only if g(a) = 0. We also say that y = a is a root or zero of a polynomial only if it is a solution to the equation g(y) = 0. Let’s consider an example to find the zeroes of the second-degree polynomial g(y) =y2+2y-15. To do this we simply solve the equation by using the factorization of quadratic equation method as shown below:

 y2+2y-15=0

0 = (y+5)(y-3) 

y = -5 and y = 3

Thus, this shows a second-degree polynomial has two roots. This second-degree polynomial                  y2+2y-15 has two zeros or roots which are – 5 and 3.

FACTOR THEOREM FORMULA:

As per a factor theorem,  (y – a) can be considered as a factor of the polynomial g(y) of degree n ≥ 1, if and only if g(a) = 0 (where a is a real number). The basic formula of the factor theorem is g(y) = (y – a) q(y). It is important to note that all the following statements given below apply for any polynomial g(y):

• (y – a) is the factor of g(y).

• g(a) = 0

• The remainder always becomes zero when g(y) is divided by (y – a).

• The solution to g(y) = 0 is a. and the zero (or root) of the function g(y) is a.

PROOF OF FACTOR THEOREM:

In order to prove the factor theorem, let us first consider a polynomial g(y) that is being divided by (y – a) ( if g(a) = 0). By using the division algorithm, the polynomial can be written as the product of its divisor and its quotient.

Dividend = (Divisor × Quotient ) + Remainder

⟹ g(y) = (y – a) q(y) + R (remainder). Here, g(y) is dividend, (y – a) is divisor, and q(y) is the quotient.

As per the remainder theorem, we get:

g(y) = (y – a) q(y) + g(a)

If we substitute g(a) = 0 then the remainder is equal to 0.

⟹ g(y) = (y – a) q(y) + 0

⟹ g(y) = (y – a) q(y)

Thus, we can say (y – a) is a factor of the polynomial g(y). we can clearly see that the factor theorem is actually a result of the remainder theorem, which states that a polynomial g(y) has a factor (y – a), if and only if a is a root of g(y) i.e., g(a) = 0.

USING THE FACTOR THEOREM TO FACTOR A THIRD-DEGREE POLYNOMIAL:

We widely use the factor theorem for second degree or quadratic polynomials. But for higher degree polynomial we use the process given below,

• Step 1: Use the synthetic division method to divide the given polynomial g(y) by the given binomial (y−a).

• Step 2: After the completion of the division, see whether the remainder is 0. If the remainder is not zero, then it means that the given binomial (y-a) is not a factor of g(y). If the remainder is zero proceed with the next step.

• Step 3: Using the division algorithm, write the given polynomial as the product of (y-a) and quadratic quotient q(y)

• Step 4: If it is possible, factorize the quadratic quotient further.

• Step 5: Express the polynomial as the product of its factors.

Using the factor theorem, let’s show that (y+2) is a factor of the polynomial  y3-6y2-y+30 and then find the remaining factors. After finding the remaining factors, we will use the factors to determine the zeros of the given polynomial.

• The first step is the synthetic division method to show that (y+2) is a factor of the third-degree polynomial y3-6y2-y+30

• After the completion of the synthetic division, we get the remainder to be zero. Hence, (y + 2) is a factor of the given polynomial y3-6y2-y+30

• Now, let’s use the division algorithm to write the given polynomial y3-6y2-y+30 as the product of the divisor (y + 2) and the quadratic quotient y2-8y+15 as (y+2)( y2-8y+1)

• Now let’s factorize the quadratic equation to write the polynomial as (y + 2)(y − 3)(y − 5).

Thus, by using the factor theorem we got the zeroes of the given polynomial y3-6y2-y+30 are    –2, 3, and 5.

IMPORTANT NOTES ON FACTOR THEOREM:

Here are a few points that should be remembered while studying the factor theorem:

• Factor theorem is mainly used to factorize the polynomials and to find the roots of that polynomial.

• In real life, factoring is useful while exchanging money, dividing any quantity into equal pieces, understanding time, and comparing prices.

• As per the factor theorem, (y – a) can be considered as the factor of polynomial g(y) of degree n ≥ 1, if and only if g(a) = 0.

CONCLUSION

Factor theorem is mainly used to factor the polynomial and to find the n roots of the polynomial. Factor theorem is very helpful for the analysis of polynomial equations. In real life, factoring can be useful, especially while exchanging money, dividing any quantity into equal pieces, understanding the time, and comparing prices. Factor theorem is a special kind of the polynomial remainder theorem that links the factor and zeros of a polynomial. The factor theorem removes all the known zeroes from a given polynomial equation and leaves all the unknown zeroes of the polynomial equation. The resultant polynomial has a lower degree in which the zeroes can be found easily.

There are two mainly used factoring methods. First is the normal method using the common factor method and the next is the method that uses the synthetic division method in order to factorize the second method is used for higher degree polynomials.