Given:
A polynomial crosses through the x-axis at -2 and touches the x-axis and turns around at 4.
To find:
The polynomial function in factored form.
Solution:
If the graph of a polynomial intersect the x-axis at [tex]x=a[/tex], then [tex](x-a)[/tex] is a factor of the polynomial.
If the graph of a polynomial touches the x-axis at [tex]x=b[/tex], then [tex](x-b)[/tex] is a factor of the polynomial with multiplicity 2. In other words [tex](x-b)^2[/tex] is the factor of the polynomial.
It is given that the polynomial crosses through the x-axis at -2. So, [tex](x+2)[/tex] is a factor of required polynomial.
It is given that the polynomial touches the x-axis and turns around at 4. So, [tex](x-4)^2[/tex] is a factor of required polynomial.
Now, the required polynomial is:
[tex]P(x)=(x+2)(x-4)^2[/tex]
Therefore, the required polynomial is [tex]P(x)=(x+2)(x-4)^2[/tex].
