what must be multiplies to x²+2x+1 to get x⁴+x³+x²+3x+2

We are given two expressions:

1. \( x^2 + 2x + 1 \)
2. \( x^4 + x^3 + x^2 + 3x + 2 \)

We need to determine what must be multiplied by \( x^2 + 2x + 1 \) to obtain \( x^4 + x^3 + x^2 + 3x + 2 \).

Let the unknown factor be \( f(x) \). Then, we can write the equation:

\[
(x^2 + 2x + 1) \cdot f(x) = x^4 + x^3 + x^2 + 3x + 2
\]

To find \( f(x) \), we can divide \( x^4 + x^3 + x^2 + 3x + 2 \) by \( x^2 + 2x + 1 \) using polynomial division.

Let me calculate this.The result of dividing \( x^4 + x^3 + x^2 + 3x + 2 \) by \( x^2 + 2x + 1 \) is \( x^2 - x + 2 \), with a remainder of 0.

Thus, the expression that must be multiplied by \( x^2 + 2x + 1 \) to get \( x^4 + x^3 + x^2 + 3x + 2 \) is:

\[
x^2 - x + 2
\]

Would you like more details on how polynomial division works or have any further questions?

Here are 5 related questions to explore:
1. How do you perform polynomial long division step-by-step?
2. What happens when the remainder is not zero in polynomial division?
3. Can you factorize \( x^2 - x + 2 \) further?
4. How do you handle division of polynomials with higher degrees?
5. How does synthetic division compare to polynomial division?

**Tip:** Polynomial division is useful when simplifying rational expressions or finding roots!

What must be multiplied to x² + 2x + 1 to get x⁴ + x³ + x² + 3x + 2?

Learn how to determine what must be multiplied by the polynomial x² + 2x + 1 to obtain x⁴ + x³ + x² + 3x + 2 using polynomial division. This guide covers algebraic concepts like polynomial division and quadratic polynomials, suitable for students in Grades 9-12.

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