Question

Help?

Answer 1

highest exponent is 3, so only 3 possible total roots

to figure out how many complex roots, first figure out how many real roots by setting the equation equal to 0, factoring it, and solving for x

Answer 2

3?

Answer 3

good, so 3 roots

[just did a quick search and "zeros over the set of complex numbers" doesn't mean complex roots, just total roots, so part A) is 3]

Answer 4

part II)

number of extrema = degree of the function - 1 = ?

Answer 5

2

Answer 6

good

for part III) what happens to the function x gets infinitely large and positive? does it diverge to +infinity or -infinity?

Answer 7

+?

Answer 8

good so a) = +infinity

what about b)? does the function approach + or -infinity as x --> -infinity?

Answer 9

-

Answer 10

-infinity?

Answer 11

good so b) -infinity

Answer 12

for part IV) use the rational root theorem

start by listing all the factors of the constant term 4, including +/- signs for each factor

Answer 13

so +1,+2,+3,+4

Answer 14

3 is not a factor of 4

so ±1, ±2, ±4

then, the leading coefficient is just 1, so its factors are just ±1

then we list all possibilities of constant factors/leading coefficient factors

Answer 15

so ±1/1, ±2/1, ±4/1, then we just simplify this to get

±1, ±2, ±4

Answer 16

for part V)

x^3 - x^2 - 4x + 4

start by splitting this into two parts

(x^3 - x^2) - (4x - 4)

then just factor out the from this and lmk what you get

(x^3 - x^2)

Answer 17

*factor out the gcf

Answer 18

x^2(x-1)

Answer 19

awesome

then the other part:

(4x - 4)

factor out the gcf again

Answer 20

4(x-1)

Answer 21

good

then we have

x^2(x-1) - 4(x-1)

we can re-combine these to get

(x^2 - 4)(x-1) {this is the opposite of the distributive property]

then, if we look at x^2 - 4 this is a difference of squares

so what does x^2 - 4 factor to?

Answer 22

(x+2)(x-2)

Answer 23

awesome so the entire factored form is (x-1)(x+2)(x-2) = your ans

Answer 24

and that's it!