The zeroes of the function f(x) = 3x^6 – 30x^5 + 75x^4 can be found by factoring the equation. First, we factor out the greatest common factor, which in this case is 3x^4. This gives us:
f(x) = 3x^4(x^2 – 10x + 25)
Next, we simplify the quadratic expression. The expression x^2 – 10x + 25 can be factored as (x – 5)(x – 5) or (x – 5)². Therefore, we write:
f(x) = 3x^4(x – 5)²
Now we can find the zeroes. Setting f(x) equal to zero, we get:
3x^4(x – 5)² = 0
This means that either 3x^4 = 0 or (x – 5)² = 0.
From 3x^4 = 0, we find x = 0, and this zero has a multiplicity of 4 since it comes from x^4. From (x – 5)² = 0, we find x = 5, with a multiplicity of 2. Thus, the zeroes of the function are:
x = 0 (with multiplicity 4) and x = 5 (with multiplicity 2).
In summary, the graph of this polynomial function will touch the x-axis at x = 0 and will cross the x-axis at x = 5, reflecting the multiplicities of the zeroes. This means the zero at x = 0 represents a point where the function rebounds from the axis, while the zero at x = 5 indicates a point where the function crosses the axis.