Binomial expansion help (1 Viewer)

[hayabusaboston]

Hai Gaiz, I lost my 1st book of notes and exercises and thus have forgotten how to do questions like this:

Expand (4+x)^5 as far as the term in x^3
Hence find the coefficient of x^3 in (3-x)(4+x)^5

The answer is -160. Plz explain how to do it a short way someone?

 

[SpiralFlex]

The coefficient of x cubed - figure out which terms expanded will give you it.





The rest of the terms don't matter since the first bracket isn't "lowering" the higher powers. Practice and you'll be able to do this mentally.

 

[hayabusaboston]

The coefficient of x cubed - figure out which terms expanded will give you it.





The rest of the terms don't matter since the first bracket isn't "lowering" the higher powers. Practice and you'll be able to do this mentally.

But wait, how come its 3*160x^3 then MINUS x*640x^2? Is it kinda like binomial proofs where you are finding out which combinations will give you the desired power of x?

 

[SpiralFlex]

But wait, how come its 3*160x^3 then MINUS x*640x^2? Is it kinda like binomial proofs where you are finding out which combinations will give you the desired power of x?

Red means you won't get the x cubed term therefore we ignore it. Green means you will. Add the terms together.

 

[D94]

But wait, how come its 3*160x^3 then MINUS x*640x^2? Is it kinda like binomial proofs where you are finding out which combinations will give you the desired power of x?

You're trying to find the coefficients of x³. If you were to expand that whole thing out (which you should not), you will get: 3*1024 + 3*1280x + ... + 3*x⁵ - x*1024 - x*1280x - ... - x*x⁵.

When you group the terms together of same powers of x, the coefficient for the x³ term will be (3*160 - 640).

 

[hayabusaboston]

You're trying to find the coefficients of x³. If you were to expand that whole thing out (which you should not), you will get: 3*1024 + 3*1280x + ... + 3*x⁵ - x*1024 - x*1280x - ... - x*x⁵.

When you group the terms together of same powers of x, the coefficient for the x³ term will be (3*160 - 640).

Yees It clicked yesterday. You can just use binomial combinations of powers ive found. But if there's an x^2 or higher in there then idk how to apply those, so I use your method.