Question

Which statement describes the graph of this polynomial function?

f(x)= x4+ x3 - 2x2
The graph crosses the x-axis at x = 2 and x = -1 and touches the x-axis at x = 0.
The graph touches the x-axis at x = 2 and x = -1 and crosses the x-axis at x = 0.
O The graph crosses the x-axis at x = -2 and x = 1 and touches the x-axis at x = 0.
The graph touches the x-axis at x = -2 and x = 1 and crosses the x-axis at x = 0.

Answer 1

Option (3)

Explanation:

Given function is,

f(x) = x⁴ + x³ - 2x²

= x²(x² + x - 2)

= x²(x² + 2x - x - 2)

= x²[x(x + 2) - 1(x + 2)]

= x²(x + 2)(x - 1)

So the factored form of the polynomial function is,

f(x) = x²(x + 2)(x - 1)

For x - intercepts,

F(x) = x²(x + 2)(x - 1) = 0

x = -2, 1

This function has even multiplicity = 2 at x = 0.

Therefore, graph of the function will touch the x-axis at x = 0

And at other roots x = -2, 1 has odd multiplicity = 1, so the graph will cross the x-axis.

Option (3) will be the correct option.