What are the zeroes of the function? What are their multiplicities?
function symbol= x4 – 4x3 + 3x2
(1 point)
Responses
The numbers –1 and –3 are zeroes of multiplicity 2; the number 0 is a zero of multiplicity 1.
The numbers –1 and –3 are zeroes of multiplicity 2; the number 0 is a zero of multiplicity 1.
The number 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeroes of multiplicity 1.
The number 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeroes of multiplicity 1.
The numbers 0 and 1 are zeroes of multiplicity 2; the number 3 is a zero of multiplicity 1.
The numbers 0 and 1 are zeroes of multiplicity 2; the number 3 is a zero of multiplicity 1.
The number 0 is a zero of multiplicity 2; the numbers –1 and –3 are zeroes of multiplicity 1.
The numbers –1 and –3 are zeroes of multiplicity 2; the number 0 is a zero of multiplicity 1.
The correct answer is: The numbers –1 and –3 are zeroes of multiplicity 2; the number 0 is a zero of multiplicity 1.
To find the zeroes of the function and their multiplicities, we need to analyze the given function:
function symbol = x^4 - 4x^3 + 3x^2
To find the zeroes, we need to find the values of x for which the function equals zero. In other words, we need to solve the equation:
x^4 - 4x^3 + 3x^2 = 0
To solve this equation, we can factor out common terms:
x^2(x^2 - 4x + 3) = 0
Now, we have two factorized terms:
1. x^2 = 0
This equation has a zero at x = 0, with a multiplicity of 2 (since it appears as a factor twice).
2. (x^2 - 4x + 3) = 0
To solve this quadratic equation, we can attempt to factor it further:
(x - 1)(x - 3) = 0
This equation gives us two additional zeros: x = 1 and x = 3, both with a multiplicity of 1.
Therefore, the zeroes of the function and their multiplicities are:
- The number 0 is a zero of multiplicity 2.
- The numbers 1 and 3 are zeroes of multiplicity 1.
Hence, the correct answer is:
The numbers 0 and 1 are zeroes of multiplicity 2; the number 3 is a zero of multiplicity 1.