Q. \(x^3-1\)

Q: Solve x^3-1=0.

x=1 or x^2+x+1=0. The quadratic has discriminant 1-4=-3, so x=\frac{-1\pm i\sqrt{3}}{2}.

Q: Find all real roots of x^3-1.

Only x=1. Since x^2+x+1 has no real solutions (discriminant -3).

Q: Compute the remainder of dividing x^3-1 by x-1.

By the Factor Theorem, remainder is 0 because x-1 divides x^3-1.

Q: Determine where x^3-1 is positive or negative.

Sign changes at x=1. For x 1, x^3-1>0.

Answer

\(x^3-1\) factors as a difference of cubes.

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

Final result: \((x-1)(x^2+x+1)\)

Detailed Explanation

Step 1: Rewrite the expression.

We are given the expression \(x^3 – 1\). The goal is usually to factor the expression (because this is a common type of problem).

Step 2: Recognize a difference of cubes.

Recall the factoring identity for the difference of cubes:

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

Here, match terms with \(x^3 – 1\).

Notice that \(x^3 = (x)^3\) and \(1 = (1)^3\). So we can set:

\[
a = x,\quad b = 1
\]

Step 3: Substitute into the identity.

Using \(a = x\) and \(b = 1\):

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

Step 4: Simplify inside the parentheses.

Compute \(x\cdot 1 = x\) and \(1^2 = 1\):

\[
(x – 1)\left(x^2 + x + 1\right)
\]

Final Answer (factored form):

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

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Algebra FAQ

Factor \(x^3-1\) over the reals.

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

Solve \(x^3-1=0\).

\(x=1\) or \(x^2+x+1=0\). The quadratic has discriminant \(1-4=-3\), so \(x=\frac{-1\pm i\sqrt{3}}{2}\).

Find all real roots of \(x^3-1\).

Only \(x=1\). Since \(x^2+x+1\) has no real solutions (discriminant \(-3\)).

Compute the remainder of dividing \(x^3-1\) by \(x-1\).

By the Factor Theorem, remainder is \(0\) because \(x-1\) divides \(x^3-1\).

Express \(x^3-1\) as a difference of cubes.

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

Determine where \(x^3-1\) is positive or negative.

Sign changes at \(x=1\). For \(x<1\), \(x^3-1<0\); for \(x>1\), \(x^3-1>0\).

Evaluate \(x^3-1\) at \(x=2\) and \(x=-1\).

\(2^3-1=7\). Also \((-1)^3-1=-2\).

Solve x^3−1 step-by-step.
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