Behaviour of polynomials at their zeroes

Algebra Level 3

Let $f(x) = x^3+1$ . Then, what is the behavior of the function at $(-1,0)$

Note : In the image given above the graph of a function $f(x) = (x-1)^2(x-3)$ is shown. Here, the graph of the function touches $x$ -axis at the point $(1,0)$ whereas it crosses $x$ -axis at the point $(3,0)$ . (This note has been provided to clarify the diffrence between touching and crossing the $x$ -axis, the function given in note has no relation to the question)

None of these choices It touches the

x

-axis at

(-1,0)

It crosses the

x

-axis at

(-1,0)

It touçhes and crosses the

x

-axis at

(-1,0)

It neither touches or crosses the

x

-axis at

(-1,0)

1 solution

Puneet Pinku
May 10, 2016

The given function f(x) = $x^3+1$ can be factorised as $x^3+1 = (x+1)(x^2-x+1)$ .

Now, the number of times the same zero occurs is called its multiplicity.For example, in the polynomial g(x) = $(x-3)^2(x+1)(x+7)$ th zero 3 has a multiplicity of two, -1 has a multiplicity of 1 and -7 also has a mltiplicity of 1. in our function f(x) = $(x+1)(x^2-x+1)$ , the zero -1 has a multiplicity of 1.

Now the behaviour of the graph of a function at its zero is largely determined by its multiplicity. When the mulplicity of a zero is an odd number, the graph crosses the x-axis at (x,0) where x is the zero of the polynomial.If its a even number, the graph just touches the x-axis at (x,0). Hence it our case since 1 is an odd number, the graph of the function will cross the x-axis at (-1,0) and we are done....

Please clarify the difference between the options "it touches and crosses the x axis" and "it crosses the x axis".

Arulx Z - 5 years, 1 month ago

The option which tells both touching and crossing the x axis is just a dummy answer which is not at all possible. I was just increasing the no. of options to decrease the probabity if one blindly ticks the answer. I hope you understand the meaning of crossing the axis from the question itself I have provided an image and a note with it to clarify the difference between touching and crossing the x axis.

Puneet Pinku - 5 years, 1 month ago

Behaviour of polynomials at their zeroes

1 solution

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