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

Q: Why do I need to factor out x⁶ first instead of going straight to the difference of squares?

Factoring out the greatest common factor (GCF) first is crucial! If you skip this step, you'll miss the solution x = 0. The GCF $ x^6 $ gives us one factor that equals zero.

Q: How do I know x² - 25 is a difference of squares?

Look for the pattern $ a^2 - b^2 $! Here, $ x^2 - 25 = x^2 - 5^2 $, which factors as $ (x-5)(x+5) $. Both terms must be perfect squares with a minus sign between them.

Q: Why does x⁶ = 0 only give me x = 0 and not six solutions?

Great observation! While $ x^6 = 0 $ is a sixth-degree equation, the only real number that when raised to the 6th power equals zero is x = 0. We say x = 0 has multiplicity 6.

Q: What does 'Answers a + b' mean in the multiple choice?

This means combine multiple answer choices! Choice (a) gives $ x = ±5 $ and choice (b) gives $ x = 0 $. Together, they include all three distinct solutions: -5, 0, and 5.

Polynomial Factoring with Zero-Product Property

Solve for x:

$x^8-25x^6=0$

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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 the term X to the power of 6

00:09 Take out the common factor from the parentheses

00:20 This is one solution that zeros the equation

00:25 Now let's check which solutions zero the second factor

00:29 Isolate X

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

00:38 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

$x^8-25x^6=0$

Step-by-step solution

To solve the equation $x^8 - 25x^6 = 0$ , we start by noticing that both terms share a common factor of $x^6$ . We can factor out $x^6$ from the expression:

$x^6(x^2 - 25) = 0$

According to the zero-product property, a product is zero if and only if at least one of the factors is zero. Therefore, we have two separate equations to solve:

$x^6 = 0$
$x^2 - 25 = 0$

For $x^6 = 0$ :

$x = 0$

For $x^2 - 25 = 0$ , this can be seen as a difference of squares, which factors as:

$(x - 5)(x + 5) = 0$

Again, using the zero-product property, we solve the factors:

$x - 5 = 0$ gives $x = 5$
$x + 5 = 0$ gives $x = -5$

The solutions to the equation are therefore $x = 0, x = 5,$ and $x = -5$ .

The correct answer choice is "Answers a + b", where $\pm 5$ and $0$ are included as solutions.

Final Answer

Answers a + b

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Extract greatest common factor first before applying special patterns
Technique: Factor $x^8 - 25x^6 = x^6(x^2 - 25)$ using GCF then difference of squares
Check: Substitute x = 0, 5, -5 back: $0^8 - 25(0^6) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Trying to factor x² - 25 without first factoring out x⁶
Don't jump straight to x² - 25 = (x-5)(x+5) without factoring out x⁶ first = missing the x = 0 solution! This overlooks the highest power common factor and loses solutions. Always extract the greatest common factor before applying special factoring patterns.

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 x⁶ first instead of going straight to the difference of squares?

Factoring out the greatest common factor (GCF) first is crucial! If you skip this step, you'll miss the solution x = 0. The GCF $x^6$ gives us one factor that equals zero.

How do I know x² - 25 is a difference of squares?

Look for the pattern $a^2 - b^2$ ! Here, $x^2 - 25 = x^2 - 5^2$ , which factors as $(x-5)(x+5)$ . Both terms must be perfect squares with a minus sign between them.

Why does x⁶ = 0 only give me x = 0 and not six solutions?

Great observation! While $x^6 = 0$ is a sixth-degree equation, the only real number that when raised to the 6th power equals zero is x = 0. We say x = 0 has multiplicity 6.

How can I be sure I found all the solutions?

Count the degree! This is an 8th-degree polynomial, so it has at most 8 solutions. We found x = 0 (with multiplicity 6), x = 5, and x = -5, giving us exactly 8 solutions total when counting multiplicity.

What does 'Answers a + b' mean in the multiple choice?

This means combine multiple answer choices! Choice (a) gives $x = ±5$ and choice (b) gives $x = 0$ . Together, they include all three distinct solutions: -5, 0, and 5.

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Solve the Polynomial Equation: x^8 - 25x^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 x⁶ first instead of going straight to the difference of squares?

How do I know x² - 25 is a difference of squares?

Why does x⁶ = 0 only give me x = 0 and not six solutions?

How can I be sure I found all the solutions?

What does 'Answers a + b' mean in the multiple choice?

🌟 Unlock Your Math Potential

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