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Linear factors give roots. Suppose there is some number α such that x-α is a factor of the polynomial p(x). We'll see that α must be a root of p(x).
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A root of a polynomial p(x) is a number ↵ 2 R such that p(↵) = 0.
Examples.
p
p 2 is a root of x 2 2.
Be aware: What we call a root is what others call a “real root”, to emphasize that it is both a root and a real number. Since the only numbers we will consider in this course are real numbers, clarifying that a root is a “real root” won’t be necessary.
A polynomial q(x) is a factor of the polynomial p(x) if there is a third polynomial g(x) such that p(x) = q(x)g(x).
Example. 3 x 3 x 2 + 12x 4 = (3x 1)(x 2 + 4), so 3x 1 is a factor of 3 x 3 x 2 + 12x 4. The polynomial x 2 + 4 is also a factor of 3x 3 x 2 + 12x 4.
If you divide a polynomial p(x) by another polynomial q(x), and there is no remainder, then q(x) is a factor of p(x). That’s because if there’s no
remainder, then p q((xx)) is a polynomial, and p(x) = q(x)