For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.
f(x) = (1/5)x^4(x^2 – 3)
the choice
1- 0, multiplicity 4, touches x-axis;
, multiplicity 1, crosses x-axis;
– , multiplicity 1, crosses x-axis
2- 0, multiplicity 4, touches x-axis
3- 0, multiplicity 4, crosses x-axis;
, multiplicity 1, touches x-axis;
– , multiplicity 1, touches x-axis
4- 0, multiplicity 4, crosses x-axis
I got the zeros which are +sqrt(3) or -sqrt(3)
but I can't understand how get multiplicity and how determined how it touches the axis
please help me and explain to me which one in choice is correct
Best Answer
Yes, 1 is correct. Multiplicity is taken from the exponent of the corresponding linear factor. For example, function $g(x) = (x - 4)^{7}$ has a zero at $x = 4$. Its corresponding linear factor is $(x - 4)$. Thus, since the exponent of this linear factor is $7$, we know that the multiplicity of the zero is $7$.
Zeroes with odd multiplicity cross through the $x$-axis, while zeroes with even multiplicity just touch the $x$-axis. Think of an easy example: linear functions have a zero of odd multiplicity and they cross, while quadratic functions can have a zero of even multiplicity if its vertex is just on the $x$-axis so that its barely touching. When you take calculus, you'll learn about this stuff in more detail: it has to do with tangent lines and derivatives!