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

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

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 ✓

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Algebraic Technique questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

More Questions

Common Factoring

Solve x⁷ - 5x⁶ = 0: Factoring High-Degree Polynomials

Solve the Polynomial Equation: x⁵ - 4x⁴ = 0

Solve 2x^90 - 4x^89 = 0: High-Degree Polynomial Equation

Solve the Equation: x^100 - 9x^99 = 0 Using Common Factoring

Factorization - Common Factor

Solve the Quadratic Equation: 3x^2+9x=0 Using Common Factor Method

Solve the Equation x⁶-4x⁴=0: Sixth-Degree Polynomial Problem

Solve the Equation: 8x - x⁴ = 0 | Fourth-Degree Polynomial

Solve the Exponential Equation: x^14 - x^7 = 0

Extracting Square Roots

Solve the Polynomial Equation: 16x² + x³ = 0 Step by Step

Solve the Equation: x⁶ + x⁵ = 0 Using Common Factor Method

Solve the Polynomial Equation: x^8 - 25x^6 = 0

Solve the Polynomial Equation: 12x⁴ - 3x³ = 0