Q. \( (x+1)^3 \)
Answer
Expand using \((x+1)^3=(x+1)(x+1)(x+1)\).
\[
(x+1)^3 = x^3 + 3x^2 + 3x + 1
\]
Detailed Explanation
We want to expand the expression \((x+1)^3\). Since the exponent is \(3\), we use the binomial expansion formula.
Step 1: Identify the binomial form
\((x+1)^3\) is in the form \((a+b)^3\), where
\(a = x\) and \(b = 1\).
Step 2: Use the binomial expansion formula for the cube
The formula is:
\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Step 3: Substitute \(a=x\) and \(b=1\)
Substitute into each term:
\[
(x+1)^3 = x^3 + 3x^2(1) + 3x(1^2) + 1^3
\]
Step 4: Simplify each term
Now simplify the powers of \(1\): \(1^2 = 1\) and \(1^3 = 1\).
\[
x^3 + 3x^2(1) + 3x(1) + 1
\]
So this becomes:
\[
x^3 + 3x^2 + 3x + 1
\]
Final Answer
\((x+1)^3 = x^3 + 3x^2 + 3x + 1\).
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Algebra FAQ
Expand \( (x+1)^3 \).
\( (x+1)^3 = (x+1)(x+1)(x+1) = x^3 + 3x^2 + 3x + 1 \).
What is the coefficient of \(x^2\) in \( (x+1)^3 \)?
\( (x+1)^3 = x^3 + 3x^2 + 3x + 1 \), so the coefficient of \(x^2\) is \(3\).
What is the constant term of \( (x+1)^3 \)?
Substitute \(x=0\): \( (0+1)^3 = 1 \). So the constant term is \(1\).
Use Pascal’s triangle to expand \( (x+1)^3 \).
Row for power \(3\) gives \(1,3,3,1\): \( (x+1)^3 = x^3 + 3x^2 + 3x + 1 \).
Find \( (x+1)^3 \) derivative.
Differentiate \(x^3 + 3x^2 + 3x + 1\): \( \frac{d}{dx}(x+1)^3 = 3x^2 + 6x + 3 \).
Factor \( x^3 + 3x^2 + 3x + 1 \) as a cube.
It matches \((x+1)^3\), so the factorization is \( (x+1)^3 \).
Evaluate \( (x+1)^3 \) at \(x=2\).
\( (2+1)^3 = 3^3 = 27 \).
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