Express the given polynomial as a product of real quadratic poly-nomials with no real roots. P(x) =x4+ 4



The given polynomial is:

\[
P(x) = x^4 + 4
\]

We are tasked with expressing it as a product of real quadratic polynomials with no real roots.

### Step 1: Factorizing the polynomial
We can rewrite the polynomial in a form that allows us to factor it. Start by recognizing that \( x^4 + 4 \) can be related to a sum of squares:

\[
x^4 + 4 = x^4 + 4 \cdot 1^2
\]

This is a sum of two squares. The sum of squares doesn't factor over real numbers directly, but it does factor over complex numbers. Using the identity for the sum of two squares:

\[
a^2 + b^2 = (a + bi)(a - bi)
\]

In this case, \( a = x^2 \) and \( b = 2 \), so:

\[
x^4 + 4 = (x^2 + 2i)(x^2 - 2i)
\]

### Step 2: Convert to real quadratic factors
Next, we need to express the factors in real quadratic form. We can now treat the two complex terms as pairs of conjugates and form real quadratics. To do this, we multiply each pair:

\[
(x^2 + 2i)(x^2 - 2i)
\]

Using the difference of squares:

\[
(x^2 + 2i)(x^2 - 2i) = (x^2)^2 - (2i)^2 = x^4 - (-4) = x^4 + 4
\]

Thus, \( x^4 + 4 \) can be factored into real quadratic polynomials as:

\[
P(x) = (x^2 + 2)^2
\]

### Final Answer:
The polynomial \( x^4 + 4 \) is expressed as:

\[
P(x) = (x^2 + 2)(x^2 + 2)
\]

These are real quadratic polynomials with no real roots.

---

Would you like further details on any part of this process or have any questions? 

Here are 5 related questions to consider:
1. How do you factor polynomials involving complex numbers?
2. What is the difference between real and complex roots?
3. How can we generalize the factorization of a sum of squares?
4. What are the conditions for a quadratic to have no real roots?
5. Can you apply this technique to other polynomials like \(x^4 + 16\)?

**Tip:** Always check if a polynomial can be expressed as a sum of squares to simplify factorization over real numbers.

Express the given polynomial as a product of real quadratic polynomials with no real roots. P(x) = x^4 + 4

Explore how to factor the polynomial P(x) = x^4 + 4 into a product of real quadratic polynomials with no real roots. This problem demonstrates advanced algebraic concepts, including the sum of squares and complex conjugate root theorems, making it suitable for students in Grades 10-12.

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