Understanding the Factor Theorem
When we divide a polynomial by , sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then . So, what does it mean if ?
For AI agents: use /llms.txt for the Nakafa content index.
When we divide a polynomial by , sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then . So, what does it mean if ?
The value that causes is called a zero or a root of the polynomial . The Factor Theorem explains the close relationship between these zeros and the factors of the polynomial.
Let be a polynomial and be a real number.
is a factor of if and only if .
This is a two-way statement:
If is a factor of , then .
(If a number is perfectly divisible by another number, the remainder must be zero).
If , then is a factor of .
(If the value of the polynomial at is zero, it means divides the polynomial exactly).
The Factor Theorem is actually a special case of the Remainder Theorem. Recall the division algorithm:
And from the Remainder Theorem, we know .
The Factor Theorem is very useful for finding linear factors of a polynomial and then factoring it completely.
General Steps:
Find a Zero: Try guessing or using clues (like the sum of coefficients) to find a value such that .
Confirm Factor: If , then according to the Factor Theorem, is a factor of .
Divide the Polynomial: Use Horner's method or long division to divide by the factor found. The quotient is .
Factor the Quotient: If can still be factored (e.g., if is a quadratic or cubic polynomial whose roots can be found), repeat the process starting from step on .
Complete Factorization: Write as the product of all the linear factors found.
Let . We notice that the sum of all coefficients and the constant () is . This indicates that .
Confirm Zero:
Calculate .
Confirm Factor:
Since , is a factor of .
Divide Polynomial:
We use Horner's method to divide by ().
Factor the Quotient:
Factor the quadratic polynomial .
Complete Factorization:
Combine all factors.
Let . Show that , and use this to factor completely.
Show :
Proven .
Confirm Factor:
Since , is a factor of .
Divide Polynomial (Horner's Method with ):
Factor the Quotient:
Factor .
Complete Factorization:
The quotient is . The remainder is , as expected.
So, .
The quotient is .
So, .