The Factor Theorem is a special case of the Remainder Theorem. It is a rule in algebra used to check whether a given expression is a factor of a polynomial.
Statement: If f(a) =0, then (x - a) is a factor of f(x).
If we calculate f(a) and it is 0, then the remainder is also 0, so (x - a) is a factor of f(x). We can understand this with numbers too.
For example, 20 ÷ 2 = 10 with no remainder, so 2 is a factor of 20.
Check whether (y + 5) is a factor of 2y² + 7y − 15.
First, find the value of y: y+5 = 0 ⇒ y = −5
Now substitute in the polynomial:
g(−5) = 2(−5)² + 7(−5) − 15
= 2(25) − 35 − 15
= 50 − 35 − 15
= 0We got 0, which means the remainder is 0.
And this is exactly what the Factor Theorem says:
If f(a) = 0, then (x − a) is a factor.
Hence, (y + 5) is a factor.
Proof
Let f(x) be a polynomial. When f(x) is divided by (x − a), according to the division algorithm:
Dividend = (Divisor × Quotient) + Remainder
⟹ f(x) = (x − a) q(x) + r
where q(x) is the quotient and r is the remainder.
Now, substitute x = a:
f(a) = (a − a) q(a) + r
f(a) = rSo, the remainder r = f(a).
If f(a) = 0, then r = 0. Hence,
f(x) = (x − a) q(x)
Therefore, (x − a) is a factor of f(x).
Formula
According to the Factor Theorem, for any polynomial g(y) of degree n ≥ 1, (y − a) is a factor of g(y) if and only if g(a) = 0. In this case, the polynomial can be written as:
g(y) = (y − a) q(y)
where, q(y) is the quotient polynomial.
Important Results
- (y − a) is a factor of g(y).
- g(a) = 0
- The remainder is zero when g(y) is divided by (y − a).
- a is a zero (root) of the polynomial g(y)
Key differences between the factor and remainder theorems are listed as follows:
| Remainder Theorem | Factor Theorem |
|---|---|
| According to the Remainder Theorem, for any polynomial p(x) when divided by x - a, the remainder is p(a). | According to the Factor Theorem, if (x - a) is a factor of p(x), then this is true only if f(a) = 0. |
| Remainder Theorem helps us to find the remainder of the polynomial without actually dividing it. | The Factor Theorem helps us to find the factors of the given polynomial. |
Alternate Method
The Factor Theorem can be used to factorize a polynomial by testing possible zeros and then dividing the polynomial by the corresponding linear factor. Once one factor is found, the remaining polynomial can be factorized using polynomial division and standard algebraic identities.
Steps to Follow
- Write the polynomial: Express the given polynomial as f(x).
- Find possible zeros: Test suitable integer factors of the constant term by substituting them into f(x).
- Apply the Factor Theorem: If f(a) = 0, then (x − a) is a factor of the polynomial.
- Divide the polynomial: Divide the polynomial by the identified linear factor using polynomial division or synthetic division.
- Factorize the quotient: Factorize the resulting quadratic polynomial, if possible.
- Write the final factorization: Express the original polynomial as a product of its factors.
Zero of Polynomial
A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. If f(a) = 0, then a is a zero of the polynomial f(x).
Important Points
- A zero of a polynomial is also called a root.
- If f(a) = 0, then aaa is a zero and (x − a) is a factor of the polynomial.
- A non-zero constant polynomial has no zeros.
- A polynomial of degree nnn can have at most nnn distinct real zeros.
