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 number 0 is a zero of multiplicity 2; the numbers –1 and –3 are zeroes of multiplicity 1.
The numbers 1 and 3 are the zeroes of the function, and they have a multiplicity of 1. The number 0 is also a zero of the function and it has a multiplicity of 2.
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, we need to solve the equation x^4 - 4x^3 + 3x^2 = 0.
To do this, we can factor the equation by grouping. Rearranging the terms, we get:
x^4 - 4x^3 + 3x^2 = x^3(x - 4) + 3x^2 = x^2(x^2 - 4x + 3).
Now we can factor the quadratic expression inside the parentheses:
x^2 - 4x + 3 = (x - 1)(x - 3).
So the equation becomes:
x^2(x - 1)(x - 3) = 0.
Now, we set each factor equal to zero and solve for x:
x = 0, x - 1 = 0 (x = 1), and x - 3 = 0 (x = 3).
Therefore, the zeroes of the function are 0, 1, and 3.
To determine their multiplicities, we need to look at the power of each factor. The power of x - 1 is 1, the power of x - 3 is 1, and the power of x - 0 is 2 (since it appears twice as a factor).
So the multiplicities of the zeroes are:
- Zero 0 has a multiplicity of 2
- Zero 1 has a multiplicity of 1
- Zero 3 has a multiplicity of 1
Therefore, the correct response is:
"The number 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeroes of multiplicity 1."