Factor Theorem

In this lesson, we will delve into a significant corollary of the Polynomial Remainder Theorem: the Factor Theorem. This theorem is also known as the Ruffini's Theorem after the Italian mathematician Paolo Ruffini, who discovered it as well as the Ruffini's rule.

It states that if we substitute the opposite of the constant term of the divisor polynomial into the dividend polynomial and obtain a result equal to zero, then the divisor polynomial divides the dividend polynomial.

Thanks to the Factor Theorem, we will also derive two new notable algebraic identities in algebra: the difference of two cubes and the sum of two cubes.

Observation on the Remainder Theorem

In the previous lesson, we studied the remainder theorem which allows us to calculate the remainder of a division between polynomials without actually performing the division itself. This is valid in the case where the divisor polynomial is of the form .

Now, let's analyze the division between the polynomials and . We'll use Ruffini's rule:

The result of the division is and the remainder is . Since the remainder is zero, we can say that is divisible by . Moreover, based on the remainder theorem, we can assert that .

We can also reverse the reasoning, that is, based on the remainder theorem, since , then is divisible by .

Generalizing, we can state that:

The usefulness of this result is that we can verify if a polynomial is divisible by a polynomial directly using the remainder theorem by substituting into and checking that the result is zero.

This consequence of the remainder theorem is named the Factor theorem.

Factor Theorem

Examples

Difference of Two Cubes

By applying the Factor theorem, we can derive an important algebraic identity: the difference of two cubes.

Let's take the polynomial . We substitute into the value :

Hence, is divisible by according to the theorem. Now, let's try to divide the two polynomials using Ruffini's rule:

From which we find that the remainder is zero, as expected, while the quotient is:

Thus, we can rewrite the polynomial as:

This result is a significant algebraic identity, known as the difference of two cubes:

Sum of Two Cubes

Similarly to the case above, we can also derive the significant algebraic identity for the sum of two cubes.

Consider the polynomial . Substitute into the value :

Thus, is divisible by according to Ruffini's theorem. We divide the two polynomials using Ruffini's rule:

The remainder is zero, while the quotient is:

Thus, we can rewrite the polynomial as:

Again, in this case, we derive a significant algebraic identity, known as the sum of two cubes:

False Squares

In deriving the algebraic identities for the difference and sum of two cubes, we obtained expressions of the form:

These expressions are called false squares as they resemble the square of a binomial, but they lack the double product:

In Summary

In this lesson, we studied an important corollary of the remainder theorem, the Factor Theorem or Ruffini's theorem. This allows us to check if a polynomial is divisible by a polynomial of the type . One simply substitutes into and verifies that the result is zero.

Thanks to Ruffini's theorem, we derived two new algebraic identities, the difference of two cubes and the sum of two cubes. These allow us to expand the difference and sum of two algebraic expressions raised to the third power.