Dtermine an equation, in simplified form, for the family of cubic functions with zeros 2 and 4+-sqrt. 3.
y = (x-2)(x-4-√3)(x-4+√3)
= (x-2)(x2 - 8x + 13)
How did I get that quadratic? One way is to just multiply all the stuff out, the other is to realize that
4 ± √3 = (8 ± 2√3)/2 = (8 ± √12)
So, -b = 8
b2 - 4ac = 64 - 4ac = 12
so, 4ac = 52 = 4*13
We have a=1, so c=13
y = (x-2)(x2 - 8x + 13)
x3 - 10x2 + 29x - 26
Well, isn't this equation going to be a real "rootin' tootin'" time! Let's start by recalling that if a cubic function has zeros at x = a, b, and c, then its factored form can be written as (x-a)(x-b)(x-c).
In this case, the zeros are 2, 4 + sqrt(3), and 4 - sqrt(3). To simplify things a bit, let's call the latter two zeros A and B respectively.
So, our factored form would be (x - 2)(x - A)(x - B). Now, to find A and B, we need to determine their values based on the given information.
If 4 + sqrt(3) is a zero, then (x - (4 + sqrt(3))) = 0. Simplifying that, we get x - 4 - sqrt(3) = 0. Rearranging, we find that x - (4 - sqrt(3)) = 0. This means that 4 - sqrt(3) is our second zero, B.
Now we have our factored form: (x - 2)(x - (4 + sqrt(3)))(x - (4 - sqrt(3))).
But, we're not finished yet! Let's expand this equation and simplify it further.
(x - 2)(x - (4 + sqrt(3)))(x - (4 - sqrt(3)))
(x - 2)(x - 4 - sqrt(3))(x - 4 + sqrt(3))
(x - 2)((x - 4) - sqrt(3))((x - 4) + sqrt(3))
(x - 2)((x - 4)^2 - (sqrt(3))^2)
(x - 2)(x^2 - 8x + 16 - 3)
(x - 2)(x^2 - 8x + 13)
And there you have it—an equation for the family of cubic functions with zeros 2, 4 + sqrt(3), and 4 - sqrt(3): f(x) = (x - 2)(x^2 - 8x + 13).
To determine the equation for a cubic function with zeros at 2, 4 + √3, and 4 - √3, we can use the factored form of a cubic equation.
The factored form of a cubic function is given by:
f(x) = a(x - r1)(x - r2)(x - r3)
where r1, r2, and r3 are the zeros of the function.
Given that the zeros are 2, 4 + √3, and 4 - √3, we substitute these values into the factored form:
f(x) = a(x - 2)(x - (4 + √3))(x - (4 - √3))
Now, let's simplify this equation further:
f(x) = a(x - 2)(x - 4 - √3)(x - 4 + √3)
Using the difference of squares, we can simplify the right side of the equation:
f(x) = a(x - 2)[(x - 4)^2 - (√3)^2]
Simplifying further:
f(x) = a(x - 2)[(x - 4)^2 - 3]
Expanding the square and distributing a:
f(x) = a(x - 2)(x^2 - 8x + 16 - 3)
f(x) = a(x - 2)(x^2 - 8x + 13)
Expanding further:
f(x) = a(x^3 - 8x^2 + 13x - 2x^2 + 16x - 26)
f(x) = a(x^3 - 10x^2 + 29x - 26)
Therefore, the equation for the family of cubic functions with zeros 2, 4 + √3, and 4 - √3 is f(x) = x^3 - 10x^2 + 29x - 26.
To determine the equation for a cubic function with given zeros, we can start by using the zero-product property. The zero-product property states that if a polynomial function has a zero at a particular value, then the corresponding binomial factor is equal to zero.
Given the zeros 2 and 4 ± √3, we can set up the following binomial factors:
(x - 2) = 0
(x - (4 + √3)) = 0
(x - (4 - √3)) = 0
To find the equation, we can start by multiplying these factors together:
(x - 2)(x - (4 + √3))(x - (4 - √3)) = 0
Next, we simplify this expression. We can multiply the binomials using the distributive property:
(x - 2)((x - 4) - √3)((x - 4) + √3) = 0
(x - 2)((x - 4)^2 - (√3)^2) = 0
(x - 2)((x - 4)^2 - 3) = 0
(x - 2)(x^2 - 8x + 16 - 3) = 0
(x - 2)(x^2 - 8x + 13) = 0
Expanding the expression:
x^3 - 8x^2 + 13x - 2x^2 +16x - 26 = 0
x^3 - 10x^2 + 29x - 26 = 0
Therefore, the equation, in simplified form, for the family of cubic functions with zeros 2 and 4 ± √3 is:
f(x) = x^3 - 10x^2 + 29x - 26