Q. \( (x+h)^3 \)

Q: Expand (x+h)^3 .

(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 .

Q: What is the general binomial expansion for (a+b)^3 ?

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 .

Q: How do you expand (x+h)^3 using FOIL-style multiplication?

Multiply (x+h)^2 = x^2+2xh+h^2 , then (x^2+2xh+h^2)(x+h) to get x^3+3x^2h+3xh^2+h^3 .

Q: What are the coefficients of x^2h and xh^2 in (x+h)^3 ?

Coefficient of x^2h is 3. Coefficient of xh^2 is 3.

Q: Can you expand (x-h)^3 for comparison?

(x-h)^3 = x^3 - 3x^2h + 3xh^2 - h^3 .

Q: What is the derivative of (x+h)^3 with respect to x ?

\frac{d}{dx}(x+h)^3 = 3(x+h)^2 .

Q: If a term were (x+h)^3 = x^3 + \cdots , what is the constant term (with respect to x )?

Treating h as constant, the constant term is h^3 .

Answer

Use the binomial expansion:

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3.
\]

Detailed Explanation

We want to expand the expression \( (x+h)^3 \). This means we will multiply \( (x+h) \) by itself three times, or use the binomial expansion formula.

Step 1: Recall the binomial expansion for a cube

The formula for expanding \( (a+b)^3 \) is:

\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]

Step 2: Identify the parts

In our problem, \( a = x \) and \( b = h \). Substitute these into the formula.

Step 3: Substitute into the expansion

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]

Step 4: Final answer

So the expanded form is:

\[
(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
\]

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Algebra FAQ

Expand \( (x+h)^3 \).

\( (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \).

What is the general binomial expansion for \( (a+b)^3 \)?

\( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).

How do you expand \( (x+h)^3 \) using FOIL-style multiplication?

Multiply \( (x+h)^2 = x^2+2xh+h^2 \), then \( (x^2+2xh+h^2)(x+h) \) to get \( x^3+3x^2h+3xh^2+h^3 \).

What are the coefficients of \( x^2h \) and \( xh^2 \) in \( (x+h)^3 \)?

Coefficient of \( x^2h \) is \(3\). Coefficient of \( xh^2 \) is \(3\).

Can you expand \( (x-h)^3 \) for comparison?

\( (x-h)^3 = x^3 - 3x^2h + 3xh^2 - h^3 \).

What is the derivative of \( (x+h)^3 \) with respect to \( x \)?

\( \frac{d}{dx}(x+h)^3 = 3(x+h)^2 \).

If a term were \( (x+h)^3 = x^3 + \cdots \), what is the constant term (with respect to \( x \))?

Treating \( h \) as constant, the constant term is \( h^3 \).

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