What are the zeroes of the function? What are their multiplicities?
function symbol= x^4 – 4x^3 + 3x^2
(1 point)
Responses
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 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 numbers 0 is a zero of multiplicity 2; the numbers -1 and -3 are zeroes of multiplicity 1.
To find the zeroes of the function x^4 – 4x^3 + 3x^2, we set the function equal to zero and solve for x.
x^4 – 4x^3 + 3x^2 = 0
To make it easier to solve, we can factor out an x^2:
x^2(x^2 – 4x + 3) = 0
Now we have two factors: x^2 = 0 and (x^2 – 4x + 3) = 0.
For the first factor, x^2 = 0, we can take the square root of both sides:
x = 0
So 0 is a zero of the function with a multiplicity of 2.
For the second factor, (x^2 – 4x + 3) = 0, we can solve it by factoring or using the quadratic formula. However, let's just go straight to the solutions:
x^2 – 4x + 3 = 0
Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/2a
In this case, a = 1, b = -4, and c = 3.
x = (-(-4) ± √((-4)^2 - 4(1)(3)))/2(1)
= (4 ± √(16 - 12))/2
= (4 ± √4)/2
= (4 ± 2)/2
The two solutions are:
x = (4 + 2)/2 = 6/2 = 3
x = (4 - 2)/2 = 2/2 = 1
So, the function has two other zeroes: 1 and 3, both with multiplicity 1.
Therefore, the correct answer is:
The numbers 0 is a zero of multiplicity 2; the numbers 1 and 3 are zeroes of multiplicity 1.