Final answer:
To determine the multiplicity of the root x = 2 for the polynomial y = x³ − 3x² + 4, we substitute x with 2 and find that the polynomial equals zero, confirming that x = 2 is a root. Since the polynomial cannot be factored further with integer coefficients, the multiplicity of the root x = 2 is 1.
Step-by-step explanation:
The question asks about the multiplicity of the root x = 2 for the polynomial y = x³ − 3x² + 4. To find the multiplicity of a root, one has to factor the polynomial and determine how many times the root appears as a factor. If a root, say x = a, appears k times in the factored form of the polynomial (e.g., (x - a)ⁿ), then the root has a multiplicity of k. In this case, we have to factor the polynomial to see if (x - 2) is a factor and if it is, how many times it appears.
To check if x = 2 is a root, we substitute x with 2 in the given polynomial:
y = (2)³ − 3(2)² + 4 = 8 - 12 + 4 = 0
Since y = 0, x = 2 is indeed a root of the polynomial. To find the multiplicity, we would normally factor the polynomial completely, but this polynomial cannot be factored with integer coefficients, indicating that x = 2 is a root once, giving it a multiplicity of 1.