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x^4+2x^3−16x^2−2x+15

Note  that we can use a special  "trick" to write this in a slightly different manner

This will make the polynomial easily "factorable"

[ P.S.  - This does not always work...but it just happens to, here  ]

(x^4 + 2x^3 - 15x^2) - ( x^2 + 2x -15)    factor

x^2 ( x^2 + 2x - 15) - 1 ( x^2 + 2x - 15)   

(x^2 - 1) ( x^2 + 2x - 15)

(x + 1) ( x - 1) ( x + 5) ( x - 3)

Setting each factor to 0  and solving for x  gives the roots  (zeros)

x =  - 1 ,  x  = 1  ,   x = -5     x  =  3

Thanks! I need help on this quesiton

Factor the polynomial function over the complex numbers.

f(x)=x4−x3−2x−4

Enter your answer in the box.

f(x) = 

Here is a graphical solution:

Factor the polynomial function over the complex numbers.

f(x)=x4−x3−2x−4

Enter your answer in the box.

f(x) = 

(in this question the answer probably have the symbol i in it)

Thank you so much! can you answer one more question for me?

f(x)=x^3+5x^2+6x

Let's examine each statement

First one

When  x  = 2      f(2)  =  (2)^3 + 5(2)^2 + 6(2)   =  8 + 20 + 12  =  40

So....this one is false

Second one

x^3 + 5x^2 + 6x     factor

x( x^2 + 5x + 6)

x(x + 3) ( x + 2)     

So....this one is true

Third one

This is false...it will only have a remainder of 0  if divided by one of its factors....and (x - 2) is not a factor

Last one

f(-2)   =  0  is true.....since  ( x + 2)  is a factor, - 2  is a root  [ it will produce a 0  when plugged into the function  and evaluated  ]

n =110 = 2 x 5 x 11
a=2, b=5, c=11
110^((3*2 + 5)/11)=110^(11/11)=110
GCD [2, 5, 11] =1
a + b + c =2 + 5 + 11 = 18

Note: With 2 as the first factor, many n will meet the restrictions given in the question.

Examples: 2, 7, 13 =182,  2, 11, 17 =374,  2, 13, 19 =494,   2, 17, 23 =782,  2, 23, 29 =1,334......etc.

NaCl    weighs  Na + Cl    = 22.989 +35.45 = 58.439 gm/mole

22.989/58.439 is the fraction tht is Na

22.989/58.439  *  x = 2.3 gm

x= 5.847 gm of Na Cl  (Table salt)

Oops....never mind.   I see you answered your posted question in the body of your submission......

Very nice, EP.....I didn't know you were so well-versed in Chemistry!!!

[ I used to be able to do this stuff in High School, but....I've slept since then......]

Yes, I can! I just did another one that I have been stuck on and I finally got the correct answer!

You're welcome !   Hope you can figure them out now.....

wow..Melody....yes, it's clearer thank you...I have to admit, I'm going to have to sit with this for a little more, but I'm sure the light will come on!!!... ..Thank you very much...and to you also Rom..Have a great weekend!!

You are fantastic!! Thank you so very much for your help! I truly appreciate it. Have a wonderful day/night!

Third one:

(realize that oxygen gas is O2   not elemental oxygen O )

Propane gas

C3 H8  has a mole weight of     C x 3 + H x 8  =   12.011 x 3 + 1.008  x 8 = 44.097 gm/mole

   155 gm / 44.097 gm/mole =3.515 moles

  From the equation EACH mole of C3H8 requires FIVE moles of oygen gas

      3.515 x 5 = 17.575 moles of O2 required

Second one:

Oxygen mole weight 15.999 gm/mole           96 gm / 15.999 gm/mole = 6 moles of O

    from the equation you are using O2 so there is only THREE moles of O2

        each mole of O2 produces TWO moles of MgO from the equation

                          so you will get SIX moles of MgO

First one:

ONE mole of Cu is required for TWO moles of HNO3

You have  .150 mole of HNO3     so you need  .075 mole of Cu (copper) (half as much)

   Copper mole weight 63.546 gm/mole

63.546 gm/mole x .075 mole = 4.766 gm of Cu required

Using brute force I come up with

(p,q) = (7,3), (11,3)

I just found two, p=3 and  p=7 both work. There may well be many more.

\(x^2 - 2px + (p^2 -5p - 1) = 0\\ x=\frac{2p\pm\sqrt{4p^2-4(p^2-5p-1)}}{2}\\ x=\frac{2p\pm\sqrt{20p+4}}{2}\\ x=\frac{2p\pm2\sqrt{5p+1}}{2}\\ x=p\pm\sqrt{5p+1}\\ \text{since x must have 2 integer solutions, 5p+1 must be a perfect square.} \)

If p=3  then 5p+1=16 which is a perfect square.

If p=7 then  5p+1=36 which is a perfect square.

Thank you! 

\(h(x)=x^2-5x\\ \)

I'll use a different concrete example to start with

\((x-6)^2=x^2-2*6x+36\\ (x-\frac{12}{2})^2=x^2-12x+(\frac{12}{2})^2\\ x^2-12x+(\frac{12}{2})^2=(x-\frac{12}{2})^2\\~\\ \text{So for your question}\\ x^2-5x+(\frac{5}{2})^2=(x-\frac{5}{2})^2\\ \text{which means}\\ x^2-5x+(\frac{5}{2})^2-(\frac{5}{2})^2=(x-\frac{5}{2})^2-(\frac{5}{2})^2\\ x^2-5x=(x-\frac{5}{2})^2-(\frac{5}{2})^2\\ x^2-5x=(x-\frac{5}{2})^2-(\frac{25}{4})\\ \)

So here, both Rom and I have 'completed the square'  

Is that any clearer?

I don’t know if I am good but I can help if I can

Hi Rom,

but why \(25 \over 4\)?

or \(5 \over2\) in the first place?..I just don't get this..