Solve the Polynomial Equation: x^10 - 16x^6 = 0

Q: Why do I need to factor out x^6 first?

Factoring out the greatest common factor simplifies the equation! Without it, you'd have to work with a 10th degree polynomial directly, which is much harder than solving two simpler equations.

Q: How do I solve x^4 = 16?

Take the fourth root of both sides. Since $ \sqrt[4]{16} = 2 $, and fourth roots can be positive or negative, you get x = ±2.

Q: Why does x^6 = 0 only give x = 0?

When you have x^6 = 0, the only number that gives zero when raised to any positive power is zero itself. So x = 0 is the only solution to this part.

Q: What is the zero product property?

If A × B = 0, then either A = 0 or B = 0 (or both). This lets you split $ x^6(x^4 - 16) = 0 $ into two separate, easier equations!

Q: How can I check my answers?

For x = 0: $ 0^{10} - 16(0^6) = 0 - 0 = 0 $ ✓For x = 2: $ 2^{10} - 16(2^6) = 1024 - 16(64) = 1024 - 1024 = 0 $ ✓For x = -2: Same result as x = 2 since all exponents are even ✓

Polynomial Factoring with Zero Product Property

$x^{10}-16x^6=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Factor with X to the fourth power

00:09 Take out the common factor from the parentheses

00:20 This is one solution that zeros the equation

00:26 Now let's check which solutions zero the second factor

00:30 Extract the root

00:35 When extracting a root there are always 2 solutions: positive and negative

00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^{10}-16x^6=0$

Step-by-step solution

To solve the equation $x^{10} - 16x^6 = 0$ , follow these steps:

Step 1: Notice that the common factor in both terms is $x^6$ . Factor it out:

$x^6(x^4 - 16) = 0$

Step 2: Apply the zero product property, which states if a product equals zero, at least one of the factors must be zero.
Therefore, we have:

$x^6 = 0$ or $x^4 - 16 = 0$

Step 3: Solve $x^6 = 0$ :

The solution to this is $x = 0$ .

Step 4: Solve $x^4 - 16 = 0$ :

Rewrite it as $x^4 = 16$
Take the fourth root of both sides:

$x = \pm\sqrt[4]{16} = \pm2$

Thus, $x = 2$ or $x = -2$ .

Conclusion: The solutions are $x = \pm 2$ and $x = 0$ .

Therefore, the correct answer is: Answers a and c

Final Answer

Answers a and c

Key Points to Remember

Essential concepts to master this topic

Factoring: Extract common factor x^6 from both terms first
Technique: Apply zero product property: if x^6(x^4 - 16) = 0, then x^6 = 0 or x^4 - 16 = 0
Check: Verify solutions: 0^10 - 16(0^6) = 0 and 2^10 - 16(2^6) = 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check all factors for zero
Don't solve only x^4 - 16 = 0 and ignore x^6 = 0 = missing the solution x = 0! This happens when you rush through factoring. Always set each factor equal to zero and solve completely.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 2x^2 $$

FAQ

Everything you need to know about this question

Why do I need to factor out x^6 first?

Factoring out the greatest common factor simplifies the equation! Without it, you'd have to work with a 10th degree polynomial directly, which is much harder than solving two simpler equations.

How do I solve x^4 = 16?

Take the fourth root of both sides. Since $\sqrt[4]{16} = 2$ , and fourth roots can be positive or negative, you get x = ±2.

Why does x^6 = 0 only give x = 0?

When you have x^6 = 0, the only number that gives zero when raised to any positive power is zero itself. So x = 0 is the only solution to this part.

What is the zero product property?

If A × B = 0, then either A = 0 or B = 0 (or both). This lets you split $x^6(x^4 - 16) = 0$ into two separate, easier equations!

How can I check my answers?

For x = 0: $0^{10} - 16(0^6) = 0 - 0 = 0$ ✓
For x = 2: $2^{10} - 16(2^6) = 1024 - 16(64) = 1024 - 1024 = 0$ ✓
For x = -2: Same result as x = 2 since all exponents are even ✓

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Solve the Polynomial Equation: x^10 - 16x^6 = 0

Polynomial Factoring with Zero Product Property

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to factor out x^6 first?

How do I solve x^4 = 16?

Why does x^6 = 0 only give x = 0?

What is the zero product property?

How can I check my answers?

🌟 Unlock Your Math Potential

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