Answer:
Yes, I agree
Step-by-step explanation:
For the cubic function
A cubic function is represented as:
[tex]f(x)=ax^3+bx^2+cx+d[/tex]
A cubic function may have 1, 2 or 3 x intercepts. This is shown below
For 3 x intercepts
[tex]y = x^3 - x[/tex]
Equate y to 0
[tex]x^3 - x = 0[/tex]
Expand
[tex]x(x^2 - 1) = 0[/tex]
Express [tex]x^2 - 1[/tex] as difference of two squares
[tex]x(x - 1)(x +1 ) = 0[/tex]
x = 0 or 1 or -1
For 2 x intercepts
[tex]y = x^3 - x[/tex]
[tex]y =(x-5)^2)(x+7)[/tex]
Equate y to 0
[tex](x-5)^2(x+7) = 0[/tex]
Expand
[tex](x-5)(x-5)(x+7) = 0[/tex]
x= 5 or x = -7
For 1 x intercept
[tex]y = x^3[/tex]
Equate y to 0
[tex]x^3 = 0[/tex]
Take cube roots of both sides
[tex]x = 0[/tex]
It has been shown above that a cubic function may have 1, 2 or 3.
So, I agree to the statement
For the quadratic function
A quadratic function will not have any x intercept when the function can not be factorized;
E.g.
[tex]y = x^2 + x + 17[/tex]
The above function has no x intercept.
A quadratic function will have at least 1 x intercept when the function can be factorized;
E.g.
[tex]y = x^2- 6x + 9[/tex]
Equate y to 0
[tex]x^2- 6x + 9 = 0[/tex]
Expand
[tex]x^2 - 3x - 3x + 9 = 0[/tex]
[tex](x - 3)(x-3) = 0[/tex]
[tex]x = 3[/tex]
We've shown that a quadratic may have no x intercept, and it may also have x intercept(s)
Hence, I agree to both statement
