Question

Do you agree with the statement that " cubic functions must have at least 1 x-intercept but not more than 3, whereas quadratic functions may or may not have x-intercepts

Answer 1

Yes, I agree

Explanation:

For the cubic function

A cubic function is represented as:

`f(x)=ax^3+bx^2+cx+d`

A cubic function may have 1, 2 or 3 x intercepts. This is shown below

For 3 x intercepts

`y = x^3 - x`

Equate y to 0

`x^3 - x = 0`

Expand

`x(x^2 - 1) = 0`

Express
`x^2 - 1` as difference of two squares

`x(x - 1)(x +1 ) = 0`

x = 0 or 1 or -1

For 2 x intercepts

`y = x^3 - x`

`y =(x-5)^2)(x+7)`

Equate y to 0

`(x-5)^2(x+7) = 0`

Expand

`(x-5)(x-5)(x+7) = 0`

x= 5 or x = -7

For 1 x intercept

`y = x^3`

Equate y to 0

`x^3 = 0`

Take cube roots of both sides

`x = 0`

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.

`y = x^2 + x + 17`

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.

`y = x^2- 6x + 9`

Equate y to 0

`x^2- 6x + 9 = 0`

Expand

`x^2 - 3x - 3x + 9 = 0`

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

`x = 3`

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