Please help, expand f(x)=x^4 - X^3 + X^2+ X-

First option, and the expanded form is:

f(x) = x^4 - 7*x^3 + 7*x^2 + 21*x - 30

Step-by-step 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

d. vertex

step-by-step explanation: