Consider the following equation and Solve the given equation by factoring. Write your answer in reduced fraction form, if necessary.

121x4−100x2=0

The given equation is:

\[
121x^4 - 100x^2 = 0
\]

### Step 1: Factor out the common terms
We can factor this equation by first recognizing a common term \(x^2\):

\[
x^2(121x^2 - 100) = 0
\]

### Step 2: Apply the difference of squares
Now, focus on the quadratic expression inside the parentheses, \(121x^2 - 100\). This is a difference of squares, since:

\[
121x^2 - 100 = (11x)^2 - 10^2
\]

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can factor it as:

\[
(11x - 10)(11x + 10)
\]

Thus, the full factored equation becomes:

\[
x^2(11x - 10)(11x + 10) = 0
\]

### Step 3: Solve each factor
Now set each factor equal to zero and solve for \(x\):

1. \(x^2 = 0\) gives \(x = 0\)
2. \(11x - 10 = 0\) gives \(x = \frac{10}{11}\)
3. \(11x + 10 = 0\) gives \(x = \frac{-10}{11}\)

### Final Solution:
The solutions to the equation are:

\[
x = 0, \quad x = \frac{10}{11}, \quad x = \frac{-10}{11}
\]

Would you like more details on any step? Here are some related questions to explore:

1. What is the difference of squares method in factoring?
2. How can you factor higher-degree polynomials like \(x^4\)?
3. Why do we set each factor equal to zero when solving by factoring?
4. How do you solve equations with quadratic terms like \(x^2\)?
5. What are the other methods to solve polynomial equations?

**Tip**: Always check for common factors first when factoring a polynomial—it simplifies the equation!

Consider the following equation and Solve the given equation by factoring. Write your answer in reduced fraction form, if necessary.

121x^4 − 100x^2 = 0

Learn how to solve the polynomial equation 121x^4 − 100x^2 by factoring. This detailed solution covers factoring techniques, the difference of squares, and the zero product property. Perfect for high school students in Grades 9-11.

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