PUMPA - SMART LEARNING
எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்
Book Free DemoLet us derive the cubic identities with the help of known identitites.
Expansion of \((x+y)^3\):
Substitute \(a = b = c = y\) in the identity \((x+a)(x+b)(x+c)\) \(=\) \(x^3\)\(+(a+b+c)x^2\)\(+(ab+bc+ca)x+abc\).
Consider the LHS, \((x+a)(x+b)(x+c)\).
\((x+a)(x+b)(x+c)\) \(=\) \((x+y)(x+y)(x+y)\)
\(=\) \((x+y)^{3}\)
Consider the RHS, \(x^3\)\(+(a+b+c)x^2\)\(+(ab+bc+ca)x+abc\).
\(x^3\)\(+(a+b+c)x^2\)\(+(ab+bc+ca)x+abc\) \(=\) \(x^3\)\(+(y+y+y)x^2\)\(+(yy+yy+yy)x+yyy\)
\(=\) \(x^3\)\(+(3y)x^2\)\(+(y^2+y^2+y^2)x+y^3\)
\(=\) \(x^3\)\(+3yx^2\)\(+3y^2x+y^3\)
Thus, the identity is \((x+y)^3\) \(=\) \(x^3+3x^2y+3xy^2+y^3\).
The obtained cubic identity can also be rewritten as follows:
Consider the standard identity, \((x+y)^3\)\(=\)\(x^3+3x^2y\)\(+3xy^2+y^3\).
Take the factor \(3xy\) from the middle two terms of RHS.
Thus, .
Expansion of \((x-y)^3\):
Replace \(y\) by \(-y\) in the cubic identity of \((x+y)^3\)\(=\)\(x^3+3x^2y\)\(+3xy^2+y^3\).
\((x+(-y))^3\) \(=\) \(x^3+3x^2(-y)+3x(-y)^2+(-y)^3\)
\((x-y)^3\) \(=\) \(x^3-3x^2y+3xy^2-y^3\)
Thus, the identity is \((x-y)^3\) \(=\) \(x^3-3x^2y+3xy^2-y^3\)
The obtained cubic identity can also be rewritten as follows:
Consider the standard identity, \((x-y)^3\) \(=\) \(x^3-3x^2y+3xy^2-y^3\).
Take the factor \(3xy\) from the middle two terms of RHS.
Thus,
Let us summarize the identities...
- \((x+y)^3\)\(=\)\(x^3+3x^2y\)\(+3xy^2+y^3\) or
- \((x-y)^3\) \(=\) \(x^3-3x^2y+3xy^2-y^3\) or
Click here! to explore some examples on the expansion of cubic terms.