Q. \[ x^2+1 \]

Q: Factor and simplify x^2+1.

x^2+1 is irreducible over the reals and does not factor nicely with real numbers. Over complex numbers: x^2+1=(x-i)(x+i).

Q: Solve x^2+1\ge 0.

x^2\ge -1, which is true for all real x. So the solution set is \mathbb{R}.

Q: Find the integral \int (x^2+1)\,dx.

\int x^2\,dx+\int 1\,dx=\frac{x^3}{3}+x+C.

Q: Determine the minimum value of x^2+1 for real x.

Since x^2\ge 0, the minimum is 1, attained at x=0.

Answer

We can factor the expression as a quadratic:

\[
x^2 + 1
\]

It does not factor over the real numbers into linear factors (since \(x^2=-1\) has no real solutions).

Final result: \(x^2 + 1\) (irreducible over the reals).

Detailed Explanation

Let’s solve the expression “\(x^2+1\)” by understanding what it means.

Step 1: Identify the expression.

The problem gives the expression:

\[
x^2+1
\]

Step 2: Note that no equation is provided.

You only wrote an expression, not an equation such as \(x^2+1=0\). Since there is no “equals” sign, there is nothing to solve for.

Step 3: Simplify if possible.

The expression \(x^2+1\) is already in simplest polynomial form. There are no like terms to combine and no multiplication to distribute.

Final Answer:

\[
x^2+1
\]

See full solution

Graph

Need help solving x²+1? Try our AI homework tools now!

Homework Helper

Algebra FAQ

Factor and simplify \(x^2+1\).

\(x^2+1\) is irreducible over the reals and does not factor nicely with real numbers. Over complex numbers: \(x^2+1=(x-i)(x+i)\).

Find the roots of \(x^2+1=0\).

\(x^2=-1\). So \(x=\pm i\).

Solve \(x^2+1=5\).

\(x^2=4\), so \(x=\pm 2\).

Solve \(x^2+1\ge 0\).

\(x^2\ge -1\), which is true for all real \(x\). So the solution set is \(\mathbb{R}\).

Compute the derivative of \(x^2+1\).

\(\frac{d}{dx}(x^2+1)=2x\).

Find the integral \(\int (x^2+1)\,dx\).

\(\int x^2\,dx+\int 1\,dx=\frac{x^3}{3}+x+C\).

Determine the minimum value of \(x^2+1\) for real \(x\).

Since \(x^2\ge 0\), the minimum is \(1\), attained at \(x=0\).

Solve \(x^2+1\) step by step.
Use our AI tools for practice.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

AI Math Solver Geometry AI Trig Solver