Question

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♥write the identity of (x+y)³

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Answer 1

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(x+y)3=(x+y)(x+y)(x+y)

taking the first two:

(x+y)(x+y)=x(x+y)+y(x+y)

=x2+xy+yx+y2

=x2+2xy+y2.

This gives us

(x+y)3=(x2+2xy+y2)(x+y)

this can be expanded out into:

x(x2+2xy+y2)+y(x2+2xy+y2)=x3+2x2y+xy2+yx2+2xy2+y3

this can be simplified into

x3+3x2y+3xy2+y3

so (x+y)3=x3+3x2y+3xy2+y3

$\sf\huge\bold\pink{༄❁❁❁❁❁༄ }$

Answer 2

(x+ʏ)3=(x+ʏ)(x+ʏ)(x+ʏ)

ᴛᴀᴋɪɴɢ ᴛʜᴇ ғɪʀsᴛ ᴛᴡᴏ:

(x+ʏ)(x+ʏ)=x(x+ʏ)+ʏ(x+ʏ)

=x2+xʏ+ʏx+ʏ2

=x2+2xʏ+ʏ2.

Tʜɪs ɢɪᴠᴇs ᴜs

(x+ʏ)3=(x2+2xʏ+ʏ2)(x+ʏ)

ᴛʜɪs ᴄᴀɴ ʙᴇ ᴇxᴘᴀɴᴅᴇᴅ ᴏᴜᴛ ɪɴᴛᴏ:

x(x2+2xʏ+ʏ2)+ʏ(x2+2xʏ+ʏ2)=x3+2x2ʏ+xʏ2+ʏx2+2xʏ2+ʏ3

ᴛʜɪs ᴄᴀɴ ʙᴇ sɪᴍᴘʟɪғɪᴇᴅ ɪɴᴛᴏ

x3+3x2ʏ+3xʏ2+ʏ3

sᴏ (x+ʏ)3=x3+3x2ʏ+3xʏ2+ʏ3

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