https://file.aiquickdraw.com/mathbot/file/1725809302174v1u4sous.png

The image contains a set of algebraic expressions that need to be factored using algebraic identities. Let's go through each one step by step.

### Expression 1: \( n^4 - 10n^2 + 25 \)
This is a quadratic expression in terms of \( n^2 \). We can rewrite it as:
\[
n^4 - 10n^2 + 25 = (n^2 - 5)^2
\]
This is a **perfect square** identity.

### Expression 2: \( y^3 - y - 6 \)
This is a cubic expression. To factor this, we can apply trial and error with the Rational Root Theorem. After finding one factor, we can use synthetic division or long division. Upon factoring, we get:
\[
y^3 - y - 6 = (y - 2)(y^2 + 2y + 3)
\]

### Expression 3: \( y^3 + y - 6 \)
Similarly, using the Rational Root Theorem and factoring, we get:
\[
y^3 + y - 6 = (y - 2)(y^2 + 2y - 3)
\]

### Expression 4: \( 1 - (3a + z)^2 \)
This is a **difference of squares**:
\[
1 - (3a + z)^2 = (1 - (3a + z))(1 + (3a + z))
\]
This can be simplified further to:
\[
(1 - 3a - z)(1 + 3a + z)
\]

### Expression 5: \( 20x^2 + 20x + 6 \)
We can factor this quadratic expression by factoring out the common factor:
\[
20x^2 + 20x + 6 = 2(10x^2 + 10x + 3)
\]
The quadratic inside the parentheses cannot be factored further, so this is the final factorization.

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Let me know if you would like further details on any of these factorizations. Here are some follow-up questions to explore:

1. How would you factor higher-degree polynomials like \( y^4 + 4y^2 + 4 \)?
2. Can you apply the difference of squares identity to more complex expressions?
3. How does the Rational Root Theorem help in factoring cubic equations?
4. What are other common algebraic identities used in factoring polynomials?
5. How would you factor a cubic expression like \( 2x^3 - 3x^2 - 11x + 6 \)?

**Tip**: Always check for common factors before attempting other factoring methods, as it simplifies the process.

Factor the following expressions using algebraic identities: n^4 - 10n^2 + 25, y^3 - y - 6, y^3 + y - 6, 1 - (3a + z)^2, 20x^2 + 20x + 6

Learn how to factor expressions like n^4 - 10n^2 + 25 and y^3 - y - 6 using algebraic identities and the Rational Root Theorem. A step-by-step guide covering factorizations of quadratic and cubic expressions.

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