Hello! I need help with these four math problems. Thanks so much! :)

Question

1.) ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x. Find the x values where the extremes occur.

2.) On a 10 question exam each question is worth 10 points (no partial credit). How ways can you make a 70%?

3.) Write the polynomial in standard form. Then classify it by degree and number of terms. (a - 1)^4

4.) Find the degree of the polynomial below. x^2y − 3xy^3z + 5x + 7y

Answer 1

#1
we have
(x^4)/4 − 3x^3 + (23x^2)/2 - 15x
Fractions are a nuisance, so I prefer
x^4 - 12x^3 + 46x^2 - 60x
= x(x^3-12x^2+46x-60)
Cubics are tough, but a little synthetic division shows that we have
x(x-6)(x^2-6x+10)
So, we know we have at most 3 extremes for a 4th-degree polynomial, and that between any two real roots there is at least one extreme.

So, we know there is at least one between 0 and 6.

If all you have is algebra, finding the exact locations can be a chore.

What tools have you learned so far to tackle this kind of problem?

#2
there are 10C7 = 10*9*8 / 1*2*3 ways to answer 7 questions out of 10.

#3
(a-1)^4 = a^4-4a^3+6a^2-4a+1

#4. add all powers of all variables to find the degree.
In this case, the highest is 3xy^3z of degree 1+3+1=5.