write x^4-3x^2+x-1 in the form Q(x)(x+2)+R


To express \(x^4 - 3x^2 + x - 1\) in the form \(Q(x)(x+2) + R\), we need to perform polynomial division where we divide \(x^4 - 3x^2 + x - 1\) by \(x + 2\).

Let's proceed with polynomial long division:

1. **Step 1:** Divide the leading term \(x^4\) by \(x\). This gives us \(x^3\), so \(Q(x)\) starts as \(x^3\).
   - Multiply \(x^3\) by \(x + 2\), giving \(x^4 + 2x^3\).
   - Subtract \(x^4 + 2x^3\) from \(x^4 - 3x^2 + x - 1\), resulting in \(-2x^3 - 3x^2 + x - 1\).

2. **Step 2:** Divide the leading term \(-2x^3\) by \(x\), which gives \(-2x^2\).
   - Multiply \(-2x^2\) by \(x + 2\), giving \(-2x^3 - 4x^2\).
   - Subtract \(-2x^3 - 4x^2\) from \(-2x^3 - 3x^2 + x - 1\), resulting in \(x^2 + x - 1\).

3. **Step 3:** Divide the leading term \(x^2\) by \(x\), giving \(x\).
   - Multiply \(x\) by \(x + 2\), resulting in \(x^2 + 2x\).
   - Subtract \(x^2 + 2x\) from \(x^2 + x - 1\), resulting in \(-x - 1\).

4. **Step 4:** Divide the leading term \(-x\) by \(x\), giving \(-1\).
   - Multiply \(-1\) by \(x + 2\), giving \(-x - 2\).
   - Subtract \(-x - 2\) from \(-x - 1\), resulting in \(1\).

Thus, the quotient is \(Q(x) = x^3 - 2x^2 + x - 1\), and the remainder is \(R = 1\).

Therefore, we have:
\[
x^4 - 3x^2 + x - 1 = (x^3 - 2x^2 + x - 1)(x + 2) + 1
\]

Would you like any further clarifications or details?

Here are 5 related questions you might find interesting:
1. How do you perform synthetic division for polynomials?
2. What is the significance of the remainder in polynomial division?
3. How does polynomial long division relate to the Factor Theorem?
4. How can we use the division algorithm to factor polynomials?
5. How can you apply polynomial division to rational functions?

**Tip:** Polynomial division is similar to numerical division; you deal with one term at a time, focusing on the highest degree.

write x^4 - 3x^2 + x - 1 in the form Q(x)(x+2) + R

Learn how to express the polynomial x^4 - 3x^2 + x - 1 in the form Q(x)(x+2) + R with a detailed step-by-step solution. This example covers key algebraic techniques like polynomial long division, making it ideal for students in Grades 10-12.

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