Q. \[ \(x-1\)^3 \]

Q: How do I use the binomial theorem to expand (x-1)^3?

\,(a-b)^3=\;a^3-3a^2b+3ab^2-b^3. With a=x, b=1: x^3-3x^2+3x-1

Q: What are the coefficients of x^3, x^2, x, and the constant in (x-1)^3?

Coefficients: 1 for x^3, -3 for x^2, 3 for x, and -1 constant

Q: How can I expand (x-1)^3 by multiplying (x-1)(x-1)^2?

First (x-1)^2=x^2-2x+1. Then (x-1)(x^2-2x+1)=x^3-3x^2+3x-1

Q: How do I find the real root(s) of (x-1)^3?

(x-1)^3=0\Rightarrow x-1=0\Rightarrow x=1. Triple root, multiplicity 3

Answer

Use the binomial expansion: \( (a-b)^3=a^3-3a^2b+3ab^2-b^3 \). Let \(a=x\) and \(b=1\).

\[
(x-1)^3=x^3-3x^2+3x-1
\]

Detailed Explanation

We want to expand the expression \(\left(x-1\right)^3\) step by step.

Step 1: Identify the form \((a-b)^3\).

Let \(a = x\) and \(b = 1\). Then \(\left(x-1\right)^3 = \left(a-b\right)^3\).

Step 2: Use the binomial cube formula.

The expansion rule is

\[\left(a-b\right)^3 = a^3 – 3a^2b + 3ab^2 – b^3.\]

Step 3: Substitute \(a = x\) and \(b = 1\).

Substitute into the formula:

\[\left(x-1\right)^3 = x^3 – 3x^2\cdot 1 + 3x\cdot 1^2 – 1^3.\]

Step 4: Simplify each term.

Compute the powers and products of \(1\): \(1^2 = 1\) and \(1^3 = 1\). So the expression becomes

\[\left(x-1\right)^3 = x^3 – 3x^2 + 3x – 1.\]

Final Answer:

\[\left(x-1\right)^3 = x^3 – 3x^2 + 3x – 1.\]

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Algebra FAQ

What is the expanded form of \((x-1)^3\)?

\(\,(x-1)^3=x^3-3x^2+3x-1\,\)

How do I use the binomial theorem to expand \((x-1)^3\)?

\(\,(a-b)^3=\;a^3-3a^2b+3ab^2-b^3\). With \(a=x\), \(b=1\): \(x^3-3x^2+3x-1\)

What are the coefficients of \(x^3\), \(x^2\), \(x\), and the constant in \((x-1)^3\)?

Coefficients: \(1\) for \(x^3\), \(-3\) for \(x^2\), \(3\) for \(x\), and \(-1\) constant

How can I expand \((x-1)^3\) by multiplying \((x-1)(x-1)^2\)?

First \((x-1)^2=x^2-2x+1\). Then \((x-1)(x^2-2x+1)=x^3-3x^2+3x-1\)

What is the derivative of \((x-1)^3\) and its expanded form?

\(\frac{d}{dx}(x-1)^3=3(x-1)^2=3(x^2-2x+1)=3x^2-6x+3\)

How do I find the real root(s) of \((x-1)^3\)?

\((x-1)^3=0\Rightarrow x-1=0\Rightarrow x=1\). Triple root, multiplicity \(3\)

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