Question

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Which could be a root of f(x) = 0 with a multiplicity of 2?

-2

-4

2

4

Answer 1

The roots of a function are the x-intercepts. From the graph we see they are x=-2 and x=4. So, this eliminates -4 and 2 as possible answers. Now, do you know what "multiplicity of 2" means?

Answer 2

no

Answer 3

what does it mean?

Answer 4

ok, it just refers to the number of time the associated factor of the polynomial appears. For example:
\[f(x)=(x-1)^2\]The root of this polynomial is x=-1, and since the factor appears twice, it has a multiplicity of 2

Answer 5

oh ok

Answer 6

The point of multiplicities with respect to graphing is that any factors that occur an even number of time (twice, four times, six times, etc) are squares, so they don't change sign. Squares are always positive. This means that the x-intercept corresponding to an even-multiplicity zero can't cross the x-axis, because the zero can't cause the graph to change sign from positive (above the x-axis) to negative (below the x-axis), or vice versa. The practical upshot is that an even-multiplicity zero makes the graph just barely touch the x-axis, and then turns it back around the way it came.

Answer 7

This should be enough to answer this one :)

Answer 8

so its 4?

Answer 9

No, it's x=-2. The graph approaches the x axis and then just touches it and turns back the way it came. This can only happen for a root with a multiplicity which is an even number (such as 2).

Answer 10

ok thanks im starting to get it

Answer 11

no prob