Question

Compare the functions shown below:



f(x) = x3 + 2x2 − 4x − 3
g(x)

polynomial graph with x intercepts at negative 2, negative 1, 1, 3, y intercept at 6
h(x)

trig graph with points at 0, 3 and pi over 2, 0 and pi, negative 3 and 3 pi over 2, 0 and 2 pi, 3
Over the interval x = 0 to x = 2π


Which function has the most x-intercepts?

Answer 1

The function f(x) = x^3 + 2x^2 - 4x - 3 is a polynomial function of degree 3 and hence will have three solutions and hence 3 x-intercepts. The function g(x) has x-intercepts at x = -2, x = -1, x = 1 and x = 3. Hence function g(x) has 4 x-intercepts. The function h(x) has x-intercepts at points (pi over 2, 0) and (3 pi over 2, 0). Hence function h(x) has 2 x-intercepts. Therefore, the function with the most x-intercepts is function g(x) with 4 x-intercepts.

Answer 2

f(x) has at most 3 x-intercepts and g(x) has 4, so g(x) has more x-intercepts.

Which function has more x-intercepts?

Remember that for a polynomial p(x) of degree N, the maximum number of x-intercepts that the polynomial can have is N.

In this case, the functions are:

$f(x) = x^3 + 2x^2 - 4x - 3$

So the degree of f(x) is 3, so it has at maximum 3 x-intercepts.

The other function g(x) has the x-intercepts:

x = -2, x = -1, x = 1, x = 3

So it has 4 x-intercepts.

Then we can conclude that g(x) has more x-intercepts.

If you want to learn more about polynomials:

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