Write the simplest polynomial function with the zeros 2-i, square root of 5, and -2.
I know that the conjugates are 2+i and -the square root of five, but when I multiply it comes out wrong.
not sure
A toy rocket is launched from the ground level with an initial vertical velocity of 96ft/s. After how many seconds will the rocket hit the ground?
In order to have your "simple polynomial function" I assume you want only rational coefficients.
For that to happen the zeros of 2-i and √5 must indeed have conjugate partner factors.
to get your given zeros included, the function must be
f(x) = (x-2-i)(x-2+i)(x^2 - 5)(x + 2)
I will leave the expansion up to you.
Hint: the first two factors expanded should have given you x^2 - 4x + 5
I don't usually work with non-metric units but I recall that the equation that would describe the height of this toy rocket would be
h = -16t^2 + 96t where t is seconds, and h is the height in feet
you want that height to be zero, so
0 = -16t^2 + 96t
16t^2 - 96t = 0
t(16t - 96)=0
so t=0 or t=6
the t=0 of course gives you the initial height, and
t=6 seconds would be the answer to your question.
Thank you so much for taking the time to help me with this problem. It was so nice of you!
I thought that how you told me is the way to set it up, but the mult. of the factors is what gives me trouble. In the answer, should there be square roots and "i"s? In your function I think you left out x^+5, is that right? I need more help!!)
No problem! Yes, the function should include square roots and "i"s. The x^2 - 5 factor is included in the function I gave you, it is just not written out explicitly. The expanded form of the function should look like this:
f(x) = (x-2-i)(x-2+i)(x^2 - 5)(x + 2)
= x^4 - 4x^3 - 10x^2 + 20x + 25
I apologize for the confusion earlier. You are correct, the square root of 5 factor was missing in the function. The correct polynomial function with the given zeros is:
f(x) = (x - (2 - i))(x - (2 + i))(x - √5)(x + 2)
Now, let's expand this function step by step:
Step 1: Expand the factors involving the complex conjugate:
(x - (2 - i))(x - (2 + i)) = (x - 2 + i)(x - 2 - i)
= (x - 2)^2 - i^2
= (x - 2)^2 - (-1)
= (x - 2)^2 + 1
= x^2 - 4x + 4 + 1
= x^2 - 4x + 5
Step 2: Expand the remaining factors:
(x - √5)(x + 2) = x^2 + 2x - √5x - 2√5
= x^2 + (2 - √5)x - 2√5
Step 3: Multiply the two results from Step 1 and Step 2:
f(x) = (x^2 - 4x + 5)(x^2 + (2 - √5)x - 2√5)
Now, you can simplify or expand further if needed.
I apologize for the confusion. Let me clarify the polynomial function with the given zeros.
The zeros are 2-i, √5, and -2. To find the polynomial function, we need to consider the conjugate pairs and factor them accordingly. The conjugate pair of 2-i is 2+i, and the conjugate pair of √5 is -√5.
So, the factors of the polynomial function will be (x - (2-i)), (x - (2+i)), (x - √5), and (x + 2).
To simplify the expression, we can multiply the factors using the distributive property:
(x - (2-i))(x - (2+i))(x - √5)(x + 2)
Let's start by expanding the first two factors:
(x - (2-i))(x - (2+i)) = x^2 - (2-i)x - (2+i)x + (2-i)(2+i)
Using the FOIL method, we can multiply the last two terms:
(x - (2-i))(x - (2+i)) = x^2 - (2x+ix)-(2x-ix) + (2-i)(2+i)
= x^2 - (4x) + [(ix) + (-ix)] + (4-1i^2)
= x^2 - 4x + 4 - (i^2)
= x^2 - 4x + 4 + 1 (since i^2 = -1)
= x^2 - 4x + 5
Now, we can multiply this with the remaining two factors:
(x^2 - 4x + 5)(x - √5)(x + 2)
To simplify the expression, you can use the distributive property and multiply each term:
(x^2 - 4x + 5)(x - √5)(x + 2) = x^3 + 2x^2 - 4x^2√5 - 8x + 5x - 10√5 + 10
Combining like terms, we get:
x^3 - 2x^2 - 3x - (4x^2√5 + 8x√5 - 10√5) + 10
So, the polynomial function that satisfies the given zeros is:
f(x) = x^3 - 2x^2 - 3x - (4x^2√5 + 8x√5 - 10√5) + 10