Question

A polynomial function f(x) has a zero of 3 with multiplicity 2. (1)since the zero is 3, the graph crosses the y-axis at 3? (2) since the zero is 3, the graph goes up to the right? (3) since the multiplicity is 2, the graph crosses the x-axis? (4) since the multiplicity is 2, the graph touches but does not cross the x-axis? Please help me with this!!!

Answer 1

A polynomial function f(x) has a zero of 3 with multiplicity 2. The phrase "a function has a zero of a with a multiplicity of b" it means that the function, when factored, has "b" factors of (x-a). In other words, the function, when factored, has {{{(x - a)^b}}} in it. (There may be additional factors, too.) So your function, in factored form, includes {{{(x-3)^2}}} since it has a zero of 3 with a multiplicity of 2<br> (1)since the zero is 3, the graph crosses the y-axis at 3? Zeros are x-values which cause the function to have a value of zero. So zeros are where the graph intersects the <b>x-axis</b> (where y-values are 0)<br> (2) since the zero is 3, the graph goes up to the right? Zeros of any multiplicity have little to do with this. They are simply where the graph intersects the x-axis<br> (3) since the multiplicity is 2, the graph crosses the x-axis? The graph intersects the x-axis at 3. But zeros of any even multiplicity do not cross from one side of the x-axis to the other.<br> (4) since the multiplicity is 2, the graph touches but does not cross the x-axis? True. Zeros of even multiplicity mean there is a "bump" (either a u-like shape or an upside-down u-like shape) in the graph where just the tip of the "bump" touches the x-axis at that zero.