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The Factor Theorem The Factor Theorem is a result of the Remainder Theorem, and is based on the same reasoning. If you haven't read the lesson on the Remainder Theorem, review that topic first, and then return here. Remember that, if you divide a polynomial p(x) by a factor x – a of that polynomial, then you will get a zero remainder. Let's look again at that Division Algorithm expression of the polynomial: p(x) = (x – a)q(x) + r(x) If x – a is indeed a factor of p(x), then the remainder after division by x – a will be zero. That is: p(x) = (x – a)q(x) In terms of the Remainder Theorem, this means that, if x – a is a factor of p(x), then the remainder, when we do synthetic division by x = a, will be zero. The point of the Factor Theorem is that, if you synthetic-divide a polynomial by x = a and get a zero remainder, then, not only is x = a a zero of the polynomial (courtesy of the Remainder Theorem), but x – a is also a factor (courtesy of the Factor Theorem). Just as with the Remainder Theorem, the point here is not to do long division of a given polynomial by a given factor. You already knew how to do that, and the point of this Theorem is to make your life simpler. So, when faced with a Factor Theorem problem, you will use synthetic division and check for a zero remainder. Here are some examples of how the Factor Theorem is used:
For x – 1 to be a factor of f (x) = 2x4 + 3x2 – 5x + 7, the Factor Theorem says that x = 1 must be a zero of f (x). That is, to test whether x – 1 is a factor, we set x – 1 equal to zero and solve for x = 1, and then use synthetic division to divide f (x) by x = 1. Remember to insert zeroes for the omitted powers of x in 2x4 + 3x2 – 5x + 7:
Since the remainder is not zero, then, according to the Factor Theorem: x – 1 is not a factor of f (x).
If x + 4 is a factor, then, setting this factor equal to zero, x = –4 is a root. To do the required verification, we need to check that, when we use synthetic division on f (x), with x = –4, we get a zero remainder:
The remainder is zero, so, by the Factor Theorem: x + 4 is a factor of 5x4 + 16x3 – 15x2 + 8x + 16. In practice, the Factor Theorem is used when factoring polynomials completely. Rather than trying various factors by using long division, you will use synthetic division and the Factor Theorem. That is, any time you divide by a number and get a zero remainder in the synthetic division, this means that the number is a root, so "x minus the number" is a factor. Then you will continue the division with the resulting smaller polynomial. Here's an example:
If x = –2 is a zero, then x + 2 = 0, so x + 2 is a factor. Similarly, if x = 1/3 is a zero, then x – 1/3 = 0, so x – 1/3 is a factor. So by giving me two of the zeroes, they have also given me two factors: x + 2 and x – 1/3. Since I started with a fourth-degree polynomial, then, once I factor out these two given factors, I'll be left with a quadratic, which I can solve by using the Quadratic Formula or some other method. The Factor Theorem says that I don't have to do the long division with the known factors of x + 2 and x – 1/3; instead, I can use synthetic division with the associated zeroes –2 and 1/3. Here is what I get when I do the first division with x = –2:
Since the remainder is zero (as expected), I can continue my division with the resulting factor of 3x3 – x2 + 3x – 1 (from the bottom line of the synthetic division); I will divide this by the other given zero, x = 1/3:
This leaves me with the quadratic 3x2 + 3, which I can solve: 3x2
+ 3 = 0
If the zeroes are x = –i and x = i, then the factors are x – (–i) and x – (i), or x + i and x – i. I need to remember that I divided off a "3" when I solved the quadratic; it is still part of the polynomial, and needs to be included as a factor. The fully-factored form is: 3x4 + 5x3 + x2 + 5x – 2 = 3(x + 2)(x – 1/3)(x + i)(x – i)
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