Answer:
First option, and the expanded form is:
f(x) = x^4 - 7*x^3 + 7*x^2 + 21*x - 30
Explanation:
Notice that in the answer we have the general form:
p(x) = x^4 - a*x^3 + b*x^2 + c*x - d
Where a, b, c, and d, are real positive numbers.
Knowing that we have a polynomial of degree 4, we can discard the third and fourth options, because these ones have a degree of 2 and 3.
Now we can look only at the first and second option, let's expand them and see which one is the correct option.
First option:
f(x) = (x - 2)*(x - 5)*(x - √3)*(x + √3)
Remember that:
(x^2 - a^2) = (x - a)*(x + a)
then the last two parts of the f(x) equation can be rewriten as:
(x + √3)*(x - √3) = (x^2 - √3^2) = (x^2 - 3)
Now we can rewrite the f(x) equation as:
f(x) = (x - 2)*(x - 5)*(x^2 - 3)
Now we can just expand this, we have:
f(x) = (x^2 -5*x - 2*x + 10)*(x^2 - 3)
f(x) = (x^2 - 7*x + 10)*(x^2 - 3)
f(x) = x^4 - 3*x^2 - 7*x^3 + 21*x + 10*x^2 - 30
f(x) = x^4 - 7*x^3 + 7*x^2 + 21*x - 30
We can see that this option already matches the structure we wanted:
f(x) = x^4 - a*x^3 + b*x^2 + c*x - d
(look at the signs)
So we can conclude that the correct option is the first one, and the expanded form is:
f(x) = x^4 - 7*x^3 + 7*x^2 + 21*x - 30