Question

Question #705e3

Answer 1

Factorising

Explanation:

If you factorise this polynomial, you will get:
`f(x)=x(x^2-9)`

The quadratic is a special one because it is the difference of two squares i.e. a square number minus another square number. When you spot this, you can factorise it further:

`(x^2-9)=(x+3)(x-3)`

giving you

`f(x)=x(x+3)(x-3)`

These are the three points where the graph crosses the x axis (0, -3 and 3 respectively) and if you do f(0) (or put x = 0) that gives you the y intercept which is 0 in this case.

Also remember that positive cubics have a kind of capital N shape when sketching (but it's curved :) )

Answer 2

`"see explanation"`

Explanation:

`"since this is in the geometry section I will not use calculus"`

`"find the x-intercepts (roots ) by equating to zero"`

`rArrx^3-9x=0larr" now factorise"`

`rArrx(x^2-9)=0larr x^2-9" is difference of squares"`

`rArrx(x-3)(x+3)=0`

`"equate each factor to zero"`

`x=0rArrx=0`

`x-3=0rArrx=3`

`x+3=0rArrx=-3`

`"since polynomial is of degree 3 (highest power of x ) "`
`"and has a positive leading coefficient "`

`"then graph starts down and ends up"`

`"we can choose values of x between the roots as an "`
`"indication of the shape of the graph"`

`f(-1)=-1+9=8larrcolor(red)" above x-axis"`

`f(1)=1-9=-8larrcolor(red)" below x-axis"`
graph{x^3-9x [-20, 20, -10, 10]}