What is a quartic function with only the two real zeros given?
x = -4 and x = -1
y = (x+4)(x+1)^3
y = (x+4)^2 (x+1)^2
y = (x+4)^3 (x+1)
these are my choices
A. y = x4 + 5x3 + 5x2 + 5x + 4
B. y = x4 - 5x3 - 5x2 - 5x - 4
C. y = -x4 + 5x3 + 5x2 + 5x + 4
D. y = x4 + 5x3 + 5x2 + 5x - 5
well, then you better expand my solutions to see which fits.
I got the answer as and it was right :)
A. y = x4 + 5x3 + 5x2 + 5x + 4
To find a quartic function with the given real zeros, you can start by using the zero-product property, which states that if a times b equals zero, then either a or b (or both) must equal zero.
For the given zeros x = -4 and x = -1, you can write two binomial factors as (x + 4) and (x + 1). This is because if you substitute -4 and -1 into these factors, you will get zero for both of them:
(x + 4) = 0 when x = -4
(x + 1) = 0 when x = -1
Now, you can multiply these two binomial factors to obtain a quadratic function:
(x + 4)(x + 1) = x^2 + 5x + 4
To convert this quadratic function into a quartic function, you can multiply it by another quadratic factor that has two more real zeros. This additional quadratic factor can be derived by using any two different real values that are not equal to -4 or -1. For example, you can choose -2 and 2:
(x - 2)(x + 2) = x^2 - 4
Now, multiply this additional quadratic factor by the previous quadratic function:
(x^2 - 4)(x^2 + 5x + 4) = x^4 + 5x^3 + 4x^2 - 4x^2 - 20x - 16 = x^4 + 5x^3 - 16x - 16
Therefore, the quartic function with the given real zeros x = -4 and x = -1 is:
f(x) = x^4 + 5x^3 - 16x - 16