Q. \[ (x+y)^3 \]
Answer
Expand using the binomial formula \( (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \).
\[
(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\]
Detailed Explanation
We want to expand the expression
\[
\left(x+y\right)^3
\]
Step 1: Use the binomial expansion formula for cubes.
For any numbers \(a\) and \(b\), the following identity holds:
\[
\left(a+b\right)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Step 2: Match variables with the formula.
Here, we have \(a=x\) and \(b=y\). Substitute into the formula:
\[
\left(x+y\right)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\]
Step 3: Present the final expanded form.
\[
\left(x+y\right)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\]
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Algebra FAQ
What is the expansion of \((x+y)^3\)?
\[ (x+y)^3=x^3+3x^2y+3xy^2+y^3 \]
How do I use the binomial theorem to expand \((x+y)^3\)?
\[ (x+y)^3=\sum_{k=0}^{3}\binom{3}{k}x^{3-k}y^{k}=x^3+3x^2y+3xy^2+y^3 \]
What is the coefficient of \(x^2y\) in \((x+y)^3\)?
The term \(x^2y\) corresponds to \(k=1\), so the coefficient is \(\binom{3}{1}=3\).
What is the coefficient of \(xy^2\) in \((x+y)^3\)?
The term \(xy^2\) corresponds to \(k=2\), so the coefficient is \(\binom{3}{2}=3\).
What is the constant term (independent of \(x\)) in \((x+y)^3\) if \(y\) is constant?
Setting \(x=0\) gives \((0+y)^3=y^3\), so the constant term is \(y^3\).
Can \((x+y)^3\) be factored as a product?
Over integers it is \((x+y)(x^2+2xy+y^2)\). Note \(x^2+2xy+y^2=(x+y)^2\), so it simplifies back to \((x+y)^3\).
Expand \((x-y)^3\) and compare with \((x+y)^3\).
\[ (x-y)^3=x^3-3x^2y+3xy^2-y^3 \]
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