Question

PLEASE HELP
If the following is a polynomial function, then state its degree and leading coefficient. If it is not, then state this fact.

f(x) = -16x5 - 7x4 - 6

Answer 1

in your question the leading term is \[-16x^5\]

it is the term that contains the highest power...

the degree is the power of the leading term.

the coefficient is the number associated with x

hope that helps

Answer 2

can it even be a polynomial function if the sixteen is negative?

Answer 3

it can think about the quadratic

\[y=x^2~~~ and ~~~y=-x^2\]

the negative causes a reflection in the x axis...

same applies to this

Answer 4

okay so the degree is 5 and the leading term is -16?

Answer 5

@campbell_st ?

Answer 6

that's correct

Answer 7

cool! can you help me with one more problem?

Answer 8

ok...

Answer 9

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

2, -4, and 1 + 3i

Answer 10

i know that you do like x-2 and x+4 but i dont know what to do with the imaginary numbers

Answer 11

ok.... so the easy bit

x = 2 is a zero then (x -2) is a factor

so you can find the binomial factor for x = -4

Answer 12

ok... so then the complex roots there 2 of those

\[x = 1 + 3i~~~~and ~~~~x = 1 - 3i\]

so handle them is a similar way

subtract 1 from both sides

x - 1 = 3i x - 1 = -3i

square both sides of the equation

\[(x - 1)^2 = 9i^2~~~~~and ~~~~~ (x -1)^2 = 9i^2 \]

remember (-3)^2 = 9

so now you need to know that i^2 = -1 so substituting you get

\[(x -1)^2 = -9~~~~or~~~~~(x - 1)^2 + 9 = 0\]

I'd expand and simplify it to a quadratic factor of

\[(x^2 - 2x + 10)\]

now you just need to multiply the 3 factors for the equation

Answer 13

so foil them all??

Answer 14

i got x^4-2x^2+4x-80

Answer 15

yes... so there really isn't an easy way other than just writing them in factored form... I suppose it depends on how the question is written.

Answer 16

but its not an answer choice? did i do something wrong?

Answer 17

@campbell_st

Answer 18

i found my error it should be
x^4-2x^2+36x-80

Answer 19

well I think you are missing a term in x^3

(x^2 - 2x + 10)(x^2 + 2x - 8)

so Id do

x^2(x^2 -2x + 10) + 2x(x^2 -2x + 10) - 8(x^2 -2x + 10)

and then collect like terms

Answer 20

okay thank you very much!