the polynomial p(x)=2x^4+ax^3+bx-1 when p(x) is divided by x^2-x+1 the remainder is 3x+2 find a and b

We are given the polynomial \( p(x) = 2x^4 + ax^3 + bx - 1 \), and the fact that when it is divided by \( x^2 - x + 1 \), the remainder is \( 3x + 2 \). We need to find the values of \( a \) and \( b \).

### Step 1: Division Structure
The division of \( p(x) \) by \( x^2 - x + 1 \) follows the structure:

\[
p(x) = (x^2 - x + 1)q(x) + (3x + 2)
\]

Where \( q(x) \) is the quotient, and \( 3x + 2 \) is the remainder.

### Step 2: Express the Polynomial
Write out \( p(x) = (x^2 - x + 1)q(x) + (3x + 2) \). Here, we want to match the degrees of the polynomial on both sides. Since the degree of \( p(x) \) is 4, the degree of \( q(x) \) should be 2, so assume:

\[
q(x) = px^2 + qx + r
\]

Thus, we have:

\[
p(x) = (x^2 - x + 1)(px^2 + qx + r) + (3x + 2)
\]

### Step 3: Expand the Product
Now, expand \( (x^2 - x + 1)(px^2 + qx + r) \):

\[
(x^2 - x + 1)(px^2 + qx + r) = x^2(px^2 + qx + r) - x(px^2 + qx + r) + (px^2 + qx + r)
\]

Expand each term:

1. \( x^2(px^2 + qx + r) = px^4 + qx^3 + rx^2 \)
2. \( -x(px^2 + qx + r) = -px^3 - qx^2 - rx \)
3. \( (px^2 + qx + r) = px^2 + qx + r \)

Now combine the terms:

\[
p(x) = px^4 + (q - p)x^3 + (r - q + p)x^2 + (-r + q)x + r
\]

### Step 4: Match Coefficients
Next, match the coefficients of this expansion with the terms of \( p(x) = 2x^4 + ax^3 + bx - 1 \).

From the \( x^4 \) term:
\[
p = 2
\]

From the \( x^3 \) term:
\[
q - p = a \quad \Rightarrow \quad q - 2 = a \quad \Rightarrow \quad q = a + 2
\]

From the \( x^2 \) term:
\[
r - q + p = 0 \quad \Rightarrow \quad r - (a + 2) + 2 = 0 \quad \Rightarrow \quad r = a
\]

From the \( x^1 \) term:
\[
-q + r = b \quad \Rightarrow \quad -(a + 2) + a = b \quad \Rightarrow \quad b = -2
\]

From the constant term:
\[
r = -1 \quad \Rightarrow \quad a = -1
\]

### Step 5: Final Values
Thus, we have:

\[
a = -1 \quad \text{and} \quad b = -2
\]

### Conclusion:
The values of \( a \) and \( b \) are \( \boxed{a = -1} \) and \( \boxed{b = -2} \).

---

Would you like a more detailed explanation of any part of the solution?

Here are some related questions to explore:
1. What is the quotient \( q(x) \) in this division?
2. How do we verify the remainder using synthetic or polynomial division?
3. What happens when we change the divisor to a higher-degree polynomial?
4. How would the solution differ if the remainder was a constant?
5. Can this method be extended to non-polynomial divisors?

**Tip**: Always check your result by substituting the values of \( a \) and \( b \) back into the original polynomial to verify consistency with the remainder.

The polynomial p(x) = 2x^4 + ax^3 + bx - 1 when p(x) is divided by x^2 - x + 1, the remainder is 3x + 2. Find a and b.

Learn how to solve the polynomial division problem p(x) = 2x^4 + ax^3 + bx - 1, where the remainder when divided by x^2 - x + 1 is 3x + 2. This problem covers polynomial division, the remainder theorem, and algebraic manipulation to find the unknown coefficients a and b. Suitable for Grades 10-12.

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