Linear Algebra For Dummies
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The factor theorem states that you can go back and forth between the roots of a polynomial and the factors of a polynomial. In other words, if you know one, you know the other. At times, your teacher or your textbook may ask you to factor a polynomial with a degree higher than two. If you can find its roots, you can find its factors.

In symbols, the factor theorem states that if xc is a factor of the polynomial f(x), then f(c) = 0. The variable c is a zero or a root or a solution — whatever you want to call it (the terms all mean the same thing).

Here’s an example. Say you have to look for the roots of the polynomial f(x) = 2x4 – 9x3 – 21x2 + 88x + 48. You find that they are x = –1/2, x = –3, and x = 4 (multiplicity two). How do you use those roots to find the factors of the polynomial?

The factor theorem states that if x = c is a root, (xc) is a factor. For example, look at the following roots:

  • If x = –1/2, (x – (–1/2)) is your factor, which you write as (x + 1/2).

  • If x = –3 is a root, (x – (–3)) is a factor, which you write as (x + 3).

  • If x = 4 is a root, (x – 4) is a factor with multiplicity two.

You can now factor f(x) = 2x4 – 9x3 – 21x2 + 88x + 48 to get f(x) = 2(x + 1/2)(x + 3)(x – 4)2. Observe that 2 is a factor because 2 is the leading coefficient (the coefficient of the term with the highest exponent.)

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Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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