Question

Find the third degree polynomial with real coefficients such that 5 and are zeros and
Answer
X^3+ 5x^2 + 2x – 96
4x^3 - 20x^2– 4x – 80
x^3 – 5x^2+ 4x – 20
2x^3 – 10x^2 – 8x + 40

Answer 1

"such that 5 and are zeros and"...that's just an odd phrase...

Answer 2

hmm

Answer 3

typo maybe?

Answer 4

2i and f(2)=-96 sorry

Answer 5

so it's

Find the third degree polynomial with real coefficients such that 5 and 2i are zeros and f(2) = -96 ???

Answer 6

yes

Answer 7

If x = 2i is a zero, then x = -2i is also a zero since complex zeros come in conjugate pairs

Answer 8

So we have 3 zeros: 5, 2i, -2i

Answer 9

If x = 2i and x = -2i are zeros, then


x = 2i or x = 2i

x^2 = (2i)^2 or x^2 = (-2i)^2

x^2 = 4i^2 or x^2 = 4i^2

x^2 = 4i^2

x^2 = 4(-1)

x^2 = -4

x^2 + 4 = 0

This means that x^2 + 4 is a factor

Answer 10

ok

Answer 11

next??

Answer 12

Well if 5 is a zero, then x-5 is a factor

Answer 13

So we have two factors x-5 and x^2+4

Answer 14

So we then get

k(x-5)(x^2+4)

Answer 15

ok

Answer 16

next??

Answer 17

you're told that f(2)=-96, so

f(x) = k(x-5)(x^2+4)

f(2) = k(2-5)((2)^2+4)

-96 = k(2-5)((2)^2+4)

solve for k

Answer 18

\[
f(x)=b (x^2 + 4) (x - 5)\\
f[2]= -92\\
-24b=-96\\
b=4
\]

Answer 19

wait so which equationis the answer??

Answer 20

both are the same, we just used different variables up front

Answer 21

and the order of factors is different (which doesn't matter)

Answer 22

but my question is multiple choice the choices are above within the question

Answer 23

It should be the second one if 4x is 16 x

Answer 24

the 4x^3 one??

Answer 25

\[
4 (x-5) \left(x^2+4\right)=4
x^3-20 x^2+16 x-80
\]

Answer 26

None of your choices are good.

Answer 27

Are you sure you wrote them down right

Answer 28

yes

Answer 29

The answer is none of the above

Answer 30

ok thank you could you also help me with this other one

find all the real zeros :
f(x)=x^4-2x^3+10x^2-18x+9
f(x)=3x^4+27x^2+9