Expert: Scott A Wilson - 12/12/2010

Question
How can you tell from the factored form of a polynomial function whether the function has a repeated zero?

Answer
Each factor of a polynomial indicates a place where the polynomial is 0.

For example, f(x) = x² + x + 1 can't be factored and has no zeros.

For another, f(x) = x² - 3x + 2, this is f(x) = (x-2)(x-1).
The zeros are then at x-2 = 0 and x-1 = 0.  This is x=2 and x=1,
so there are two zeros.

If f(x) = x² + 2x + 1, then f(x) = (x+1)(x+1), which is two factors, but there is still only one zero.  The zero is at x = -1.

If f(x) = (x-1)(x-1)(x+3), there would be 3 factors to set to 0,
but two would give the same result.

Basically, the way to tell by factors if one zero is repeated is if you get the same factor more than once.

This assumes the first term in the polynomial has coefficient 1, so factor it out.  For example, if you had 4x² + 20x + 16, factor out 4,
giving 4(x²+5x+4).  This factors to 4(x+1)(x+4), so the roots of the equatioin are x=-1 and x=-4.