You may have noticed a pattern in Try This 2 that mathematicians call the remainder theorem. The theorem simply states that when a polynomial is divided by the binomial x − a (where a is a particular number), the remainder will be the number obtained when a is substituted into the polynomial for x.
The formal way of stating this is when a polynomial in x, P(x), is divided by a binomial of the form x − a, the remainder will be P(a).1
Put together the following facts:
Combining these facts results in what mathematicians call the factor theorem: x − a is a factor of a polynomial in x, P(x), if and only if P(a) = 0.2
In the following textbook example, you will see how the factor theorem can be used to identify factors by determining the remainder.
Read “Link the Ideas” on page 127 and “Example 1” on page 128 of the textbook. As you read, pay special attention to the sign of a substituted into the polynomials. For example, since the factor theorem states that the divisor must be in the form of x − a (note the minus sign), if −2 is the a-value, the corresponding divisor is x − (−2), which is normally written x + 2.
Remember that the factor theorem states that the divisor must be in the form x − a (note the minus sign). The way to write x + 4 in this form is to write x − (−4); therefore, the a-value is −4.
Going back to the example you read before Try This 2, f(x) = 2x3 + 6x2 − 20x − 48, it can quickly be determined that x − 1 is not a factor without doing tedious long division.
Similarly, it can quickly be determined that x + 4 is a factor; therefore, long division could be performed to find another factor of the polynomial.
