What are the zeros of the function? what are their multiplicities
f(x)=4x^3-20x^2+24x
numbers -3,-2,and 0 are multiplicity of 1
numbers 3,2,and 0 are multiplicity of 1
thank you
Oh, those zeros are quite the characters! It seems like they have quite the personalities.
The zeros of the function are -3, -2, 0, 2, and 3. Now, let's talk about their multiplicities.
First up, we have -3 and 3. They both have a multiplicity of 1, meaning they are just being themselves, showing up once and saying, "Hey, we're here!"
Next, we have -2 and 2. They also have a multiplicity of 1, so they join the party and say, "We're here too, time to double the fun!"
Lastly, we have our very special guest, 0. This zero also has a multiplicity of 1. It's here to spice things up and make sure everyone knows that zero is not a zero when it comes to having a good time!
So, there you have it – the zeros of the function and their multiplicities, adding a touch of humor to the mathematical world!
To find the zeros of a function, we need to solve the equation f(x) = 0. In this case, the function is given as f(x) = 4x^3 - 20x^2 + 24x.
To solve for the zeros, we set f(x) equal to 0:
4x^3 - 20x^2 + 24x = 0
Next, we factor out the common factor of 4x:
4x(x^2 - 5x + 6) = 0
Now, we have a quadratic equation inside the parentheses. We can factor it as:
4x(x - 3)(x - 2) = 0
This equation tells us that either 4x = 0, x - 3 = 0, or x - 2 = 0. Solving each equation individually, we find:
1) 4x = 0: x = 0
2) x - 3 = 0: x = 3
3) x - 2 = 0: x = 2
These are the zeros of the function f(x). Now let's determine their multiplicities.
The multiplicity of a zero refers to the number of times it appears as a factor in the factored form of the function. In this case, we can see that the zero x = 0 has a multiplicity of 1, since it appears once as a factor (4x).
Similarly, the zero x = 3 has a multiplicity of 1 because it appears as a factor (x - 3).
Lastly, the zero x = 2 also has a multiplicity of 1 because it appears as a factor (x - 2).
Therefore, the zeros of the function f(x) = 4x^3 - 20x^2 + 24x are 0, 3, and 2, and their multiplicities are all 1.
When X=0, the function would be:
f(x) = 4x^3 -20x2 + 24x
0= 4x^3 -20x2 + 24x ----->divide all by x
x(4x^2 -20x + 24) =0 ------> split -20x into -12x and -8x
x(4x^2 -12x -8x + 24)
x{4x(x-3) - 8(x -3}
x(4x-8) (x-3)
x1= 0
x2= 8/4= 2
x3= 3
Common factor of
4 x ^ 3 - 20 x ^ 2 + 24 x
are 4x
4 x ^ 3 - 20 x ^ 2 + 24 x / 4
x =
x ^ 2 - 5 x + 6
OR
4 x ^ 3 - 20 x ^ 2 + 24 x = 4 x * ( x ^ 2 - 5 x + 6 )
4 x = 0 when x = 0
Solutions of equation :
x ^ 2 - 5 x + 6
are
x = 2 and x = 3
Solutions of equation :
4 x ^ 3 - 20 x ^ 2 + 24 x = 0
are
x = 0
x = 2 and
x = 3