2) P is of degree 4;(0)= 24; zeros:2,3i?

Question

2) P is of degree 4;(0)= 24; zeros:2,3i? please help me under stand asap write a polynomial function,P, in factored form and in standard form by using the given information.

Answer 1

when it sez p is of degree 4, it means that the highest exponent is 4, so u'll have a x^4 in it. the zeros give you ur factors for in factored form. basically, it means that if x was equal to either 2 or 3i, then the whole expression would equal 0. therefore, you can start setting up your polynomial in factored form.

P(x) = (x-2)(x-3i)(x+3i)(x+y)

its x-2 because if x-2 = 0. then x=2 (which is one of ur zeros) and if one of ur factors is 0 then the entire thing =0. you have both 3i and -3i as zeros because imagenary numbers always come in pairs. y is your other zero because we dont know what it is yet.

the (0)= 24 i'm not so sure about, but i think it means that if x=0, then you'll get 24 as you answer. so if you plug in 0 for x in your polynomial so far, you get

(0-2)(0-3i)(0+3i)(0+y) = 24
(-2)(-3i)(3i)(y) = 24
-2y(-9i^2) = 24
-2y(9) = 24
-18y = 24
y = -4/3

so your last zero is -4/3
therefore, your factored equation is:

P(x) = (x-2)(x+3i)(x-3i)(x+4/3)

so thats the first part of your question: the equation in factored form.

Now, for part 2, you can either use backwards synthetic division, which i can't show u on the computer because its really confusing and you need to draw lines for it, or you can FOIL it all together, which ill show you below.

(x-2)(x+4/3)(x+3i)(x-3i)
(x^2 - 2x+4/3x-8/3)(x^2-9i^2)
(x^2 -2/3x-8/3)(x^2+9)
x^4 - 2/3x^3 - 8/3x^2 + 9x^2 -18/3x - 72/3
P(x) = x^4 - 2/3x^3 + 19/3x^2 -6x - 24

yeah thats the standard form. but i really would ask your teacher about backwards synthetic division. it works so much better than FOILing it all together, which can be a real pain. hope this helped!

Answer 2

First, let's assume your polynomial has integer
coefficients.
Next, if 3i is a root, so is -3i. Thus
(x-3i)(x+3i) = x²+9 is a factor
Next f(0) = 24 = the product of the roots.
So 9*2*r = 24, where r is the remaining root.
So 18r = 24, r = 4/3.
So the remaining factors are x-2 and x-4/3.
The polynomial in factored form is
P = (x²+9)(x-2)(x-4/3)
and in standard form,
P = x^4 - 10/3 x³ + 35/3 x² -30x + 24.