Q. \(x^3+27\)

Q: Factor x^3+27 over the integers?

x^3+27=x^3+3^3=(x+3)(x^2-3x+9).

Q: What are the real roots of x^3+27=0?

Solve x=-3 from (x+3)(x^2-3x+9)=0. The quadratic x^2-3x+9 has discriminant \Delta=9-36=-27<0, so no other real roots.

Q: Does x^3+27=0 have any complex roots?

Yes. From x^2-3x+9=0, x=\frac{3\pm i\sqrt{27}}{2}=\frac{3\pm 3i\sqrt{3}}{2}.

Q: How can we rewrite x^3+27 using the sum of cubes formula?

Use a^3+b^3=(a+b)(a^2-ab+b^2). Let a=x, b=3: (x+3)(x^2-3x+9).

Q: What is the discriminant of x^2-3x+9?

For x^2-3x+9=0, \Delta=b^2-4ac=(-3)^2-4(1)(9)=9-36=-27.

Q: Solve x^3+27=1.

Rewrite x^3= -26, so x=\sqrt[3]{-26}=-\sqrt[3]{26}.

Answer

We factor the sum of cubes:

\[
x^3+27=x^3+3^3=(x+3)\left(x^2-3x+9\right).
\]

Final result: \(\left(x+3\right)\left(x^2-3x+9\right)\)

Detailed Explanation

We want to rewrite the expression \(x^3+27\) in a more useful factored form.

Step 1: Recognize a sum of cubes pattern.

Recall the identity for a sum of cubes:

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

Step 2: Match terms in \(x^3+27\) to \(a^3+b^3\).

Identify \(a^3=x^3\), so \(a=x\).

Identify \(b^3=27\). Since \(27=3^3\), we have \(b=3\).

Step 3: Substitute \(a=x\) and \(b=3\) into the identity.

\[
x^3+27=x^3+3^3=(x+3)\left(x^2-x\cdot 3+3^2\right).
\]

Step 4: Simplify inside the parentheses.

Compute each part:

\[
x^2-x\cdot 3+3^2=x^2-3x+9.
\]

Step 5: Write the final factored form.

\[
x^3+27=(x+3)\left(x^2-3x+9\right).
\]

See full solution

Graph

Stuck on x³+27? Try our AI homework help tools!

Homework Helper

Algebra FAQ

Factor \(x^3+27\) over the integers?

\(x^3+27=x^3+3^3=(x+3)(x^2-3x+9)\).

What are the real roots of \(x^3+27=0\)?

Solve \(x=-3\) from \((x+3)(x^2-3x+9)=0\). The quadratic \(x^2-3x+9\) has discriminant \(\Delta=9-36=-27<0\), so no other real roots.

Does \(x^3+27=0\) have any complex roots?

Yes. From \(x^2-3x+9=0\), \(x=\frac{3\pm i\sqrt{27}}{2}=\frac{3\pm 3i\sqrt{3}}{2}\).

How can we rewrite \(x^3+27\) using the sum of cubes formula?

Use \(a^3+b^3=(a+b)(a^2-ab+b^2)\). Let \(a=x\), \(b=3\): \((x+3)(x^2-3x+9)\).

What is the discriminant of \(x^2-3x+9\)?

For \(x^2-3x+9=0\), \(\Delta=b^2-4ac=(-3)^2-4(1)(9)=9-36=-27\).

Solve \(x^3+27=1\).

Rewrite \(x^3= -26\), so \(x=\sqrt[3]{-26}=-\sqrt[3]{26}\).

Solve \(x^3+27\) simply here.
Try our math AI helpers today.

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

Math Solver Geometry Solver Trigonometry Solver