Write 1024x^10 - 3840x^8 + ... in the form (a+b)^n

[Guest]

Write 1024x^10 - 3840x^8 + 5760x^6 - 4320x^4 + 1620x^2 - 243 in the form (a + b)^n. Explain you steps.

How do you figure out a and b? I know that since the powers are going down by 2, that affects a somehow. I have no idea how to find b, and for n its the amount of terms minus one so it would be 5.

Thanks for help

 

[Count Iblis]

Re: Write in the form (a+b)^n

Write 1024x^10- 3840x^8 + 5760x^6- 4320x^4+ 1620x^2- 243 in the form
(a+b)^n. Explain you steps.

How do you figure our a, and b? I know that since the powers are going down by 2, that affects a somehow. I have no idea how to find b, and for n its the amount of terms minus one so it would be 5.

Thanks for help

(4x^2 - 3)^5

 

[soroban]

Re: Write in the form (a+b)^n

Hello, Anna!

Write \(\displaystyle 1024x^{10}\,-\,3840x^8\,+\,5760x^6\,-\, 4320x^4\,+\,1620x^2\,-\,243\) in the form \(\displaystyle (a\,+\,b)^n.\)
Explain your steps.

You are right . . . The exponents "go down by 2's".
. . I suspect that the binomial has an \(\displaystyle x^2\) in it.

The polynomial begins with \(\displaystyle (4x^2)^5\,\) and ends with \(\displaystyle 3^5.\)
And because the polynomial has alternating signs,
. . I further suspect that the binomial is: \(\displaystyle \,4x^2\,-\,3\)

My guess is: \(\displaystyle \,(4x^2\,-\,3)^5\)

(Expand it and you'll see that I'm right.)