Factor Theorem

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

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

The factor theorem states that if f(x) is a polynomial of degree n greater than or equal to 1, and 'a' is any real number, then (x - a) is a factor of f(x) if f(a) = 0. In other words, we can say that (x - a) is a factor of f(x) if f(a) = 0. Let us now understand the meaning of some concepts related to the factor theorem.

Zero of a Polynomial

Before learning about the factor theorem, it is essential for us to know about the zero or 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 can 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 zeros 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:

y² + 2y − 15

= (y+5)(y−3)

= 0

⇒ y =−5 and y = 3

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

Factor Theorem Formula

As per the 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. Here, a is any real number. The formula of the factor theorem is g(y) = (y – a) q(y). It is important to note that all the following statements apply for any polynomial g(y):

• (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.

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

Dividend = (Divisor × Quotient ) + Remainder

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

From the remainder theorem, we get:

g(y) = (y – a) q(y) + g(a)

If we substitute g(a) = 0 then the remainder is 0,

⟹ g(y) = (y – a) q(y) + 0

⟹ g(y) = (y – a) q(y)

Thus, we can say that (y – a) is a factor of the polynomial g(y). Here we can 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 i.e., g(a) = 0.

Let's learn how to use the factor theorem with an example. Check whether (y + 5) is a factor of 2y² + 7y – 15 or not. Given that, y + 5 = 0. Then, y = - 5. Now let's substitute y = - 5 into the given polynomial equation. We get:

g(-5) = 2 (-5)² + 7(-5) – 15

= 2 (25) - 35 – 15

= 50 – 35 – 15

= 0

Thus, y + 5 is a factor of 2y² + 7y – 15.

Using the Factor Theorem To Factor a Third-Degree Polynomial

We usually use the factorization method to factor second-degree or quadratic polynomials. For higher degrees, we can use the below-given procedure to factor the polynomial:

• Step 1: Use the synthetic division of the polynomial method to divide the given polynomial g(y) by the given binomial (y−a)
• Step 2: After the completion of the division, confirm whether the remainder is 0. If the remainder is not zero, then it means that (y-a) is not a factor of g(y).
• Step 3: Using the division algorithm, write the given polynomial as the product of (y-a) and the quadratic quotient q(y)
• Step 4: If it is possible, factor the quadratic quotient further.
• Step 5: Express the given polynomial as the product of its factors.

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

• The first step is to use 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 method, we find that the remainder is zero. Hence, (y + 2) is a factor of the given polynomial.
• Now, let's use the division algorithm to write the given polynomial as the product of the divisor (y + 2) and the quadratic quotient (y²- 8y +15), that is, y³ − 6y² − y + 30 = (y+2) (y²− 8y +15).
• Let's factorize the quadratic equation y²− 8y +15 to write the polynomial as (y + 2)(y − 3)(y − 5).

Thus, by using the factor theorem, the zeros of the given polynomial y³ − 6y² − y + 30 are –2, 3, and 5.

Factor theorem and remainder theorem are similar but they refer to two different concepts. The remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros. Let's consider an example of a polynomial g(y) = y² − 2y + 1 to understand the difference:

For the given polynomial g(y), let's use the remainder theorem and put 3 as y into g(y):

g(3) = (3)² − 2(3) + 1
g(3) = 9 − 6 + 1
g(3) = 4

Hence, by the remainder theorem, the remainder when we divide y² − 2y + 1 by y−3 is 4. We can apply this in reverse too by dividing y² − 2y + 1 by y−3, and we will get the remainder equal to the value of g(3).

For the same polynomial, let's use the factor theorem g(y) = y² − 2y + 1 equals 0 when y =1.

g(1) = (1)² − 2(1) + 1
g(1) = 1 − 2 + 1
g(1) = 0

This tells us that (y−1) is a factor of y² − 2y + 1. We can also apply this in reverse. We can factor y² − 2y + 1 into (y − 1)². Thus, 1 is a zero of g(y).

Important Notes on Factor Theorem

• Factor theorem is mainly used to factor the polynomials and to find the n 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 a factor of the polynomial g(y) of degree n ≥ 1, if and only if g(a) = 0.

Related Articles

• Zeros of Quadratic Polynomial
• Factorization of Algebraic Expressions
• Polynomial Functions
• Degree of a Polynomial
• Types of Polynomials

What is Factor Theorem?

The factor theorem states that if f(x) is a polynomial of degree n greater than or equal to 1, and 'a' is any real number, then (x - a) is a factor of f(x) if f(a) = 0. It is mainly used to factor the polynomials and to find the n roots of the polynomials.

How do you Use the Factor Theorem?

We can use the below-given procedure to factor the polynomial using the factor theorem:

• Step 1: Use the synthetic division of the polynomial method to divide the given polynomial g(y) by the given binomial (y−a)
• Step 2: After the completion of the division, confirm whether the remainder is 0. If the remainder is not zero, then it means that (y-a) is not a factor of g(y).
• Step 3: Using the division algorithm, write the given polynomial as the product of (y-a) and the quadratic quotient q(y)
• Step 4: If it is possible, factor the quadratic quotient further by splitting the middle term method
• Step 5: Express the given polynomial as the product of its factors.

What is the Factor Theorem Formula?

As per the 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. Here, a is any real number. The formula of the factor theorem is g(y) = (y – a) q(y). It is important to note that all the following statements apply to any polynomial g(y):

• (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.

Explain Factor Theorem With an Example.

Let's use the factor theorem to find whether y+2 is a factor of the polynomial g(y) = y³ + 3y² + 5y + 6 or not.

g(–2) = (–2)³ + 3(–2)² + 5(–2) + 6

g(-2) = –8 + 12 – 10 + 6

g(-2) = 0

This tells us that (y+2) is a factor of y³ + 3y² + 5y + 6. We can also apply this in reverse.

What is the Importance of the Factor Theorem?

Factor theorem is mainly used to factor the polynomials and to find the n roots of that polynomial. It is a special kind of the polynomial remainder theorem that links the factors of a polynomial and its zeros. The factor theorem removes all the known zeros from a given polynomial equation and leaves all the unknown zeros. The resultant polynomial has a lower degree in which the zeros are a lot easier to find.

What is the Difference Between Factor Theorem and Remainder Theorem?

The difference between the factor theorem and the remainder theorem is that the remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros.

Where do we Use the Factor Theorem in Real Life?

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

What is the Importance of the Remainder Theorem and Factor Theorem?

Both the factor theorem and the remainder theorem come in handy to find the factors of a polynomial without using the other methods like synthetic division, long division, or any other traditional methods of factoring.

Is Factor Theorem and Remainder Theorem the Same?

No, the factor theorem and remainder theorem are not the same. While the remainder theorem relates the remainder of the division of a polynomial by a binomial with the value of a function at a point. The factor theorem relates the factors of a given polynomial to its zeros.