Question

find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value n=3;4 and 4 i are zeros; f (-1)=-170

Answer 1

Sounds like a system of equations problem.
We are looking for a 3rd degree polynomial because n = 3. So let
\[f(x) = ax^3 + bx^2 + cx + d.\]
Since there are four unknowns, we need four equations in order to solve this.

Answer 2

Since 4 is a root, you have that
\[f(4) = 0.\]
Thus equation 1 is
\[64a + 16b + 4c + d = 0\].
Also, you have that
\[f(-1) = -170.\]
So equation 2 is
\[-a+b-c+d = -170.\]

Answer 3

ok

Answer 4

so how would I put the polynomial in standard form

Answer 5

Wait, you still need two more equations!!

Answer 6

Since 4i is a root, that automatically means -4i is also a root (complex roots always travel in pairs)
So the factored form would look like
\[(x+4i)(x-4i).\]
When multiplied out this gives the polynomial
\[x^2 +16.\]
Hum...

Answer 7

Maybe we don't need the system stuff...
If we think of the polynomial in factored form
\[f(x) = a(x+r_1)(x+r_2)(x+r_3)\]
Use the fact that that 4 , 4i, and -4i are roots as above,
\[f(x) = a(x-4)(x^2+16)\]
Then we would be able to use the fact that
\[f(-1) = a(-5)(17) = -170\]
to solve for a.

Answer 8

ok

Answer 9

so a = 2.
\[f(x) = 2(x-4)(x^2+16)\]
Now all that's left to do is multiply the factors, which will give you the standard form.

Answer 10

so it 2x-8+x^2+16

Answer 11

almost, looks like you typed addition instead of multiplication.
\[(2x-8)(x^2+16)\]

Answer 12

ya my so it in standard form now rite

Answer 13

i mean my fault

Answer 14

No, you still need to multiply the factors. The goal is to have a polynomial with no parenthesis. Do you remember "FOIL"?

Answer 15

yes

Answer 16

That's what you need!
\[(2x-8)(x^2+16)\]
First, outer, inner, last
\[2x^3 + 32x - 8x^2 - 128\]
Now write in descending order
\[2x^3 -8x^2 + 32x - 128\]
To check that the answer is correct, find f(-1) as it should be -170.
done :)

Answer 17

thank you :)

Answer 18

hello