To find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] for which [tex]\( x^2 - x + 1 \)[/tex] is a factor of [tex]\( x^4 + a x^2 + b \)[/tex], we need to perform polynomial division and ensure that the remainder is zero.
### Step-by-Step Solution:
1. Polynomial Division Setup:
The polynomial [tex]\( x^4 + a x^2 + b \)[/tex] can be divided by the factor [tex]\( x^2 - x + 1 \)[/tex]. Let's perform polynomial division to find the quotient and remainder.
2. Form the Equations:
When we divide [tex]\( x^4 + a x^2 + b \)[/tex] by [tex]\( x^2 - x + 1 \)[/tex], we aim to write:
[tex]\[
x^4 + a x^2 + b = \left( x^2 - x + 1 \right) \cdot Q(x) + R(x)
\][/tex]
Where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R(x) \)[/tex] is the remainder. Since [tex]\( x^2 - x + 1 \)[/tex] is a factor, the remainder [tex]\( R(x) \)[/tex] must be zero.
3. Determine the Quotient:
Assume the form of the quotient [tex]\( Q(x) \)[/tex] is [tex]\( x^2 + px + q \)[/tex] (since division of a fourth degree polynomial by a second degree polynomial gives a second degree polynomial):
[tex]\[
\left( x^2 - x + 1 \right) \left( x^2 + px + q \right)
\][/tex]
4. Multiply and Simplify:
Expand the product:
[tex]\[
(x^2 - x + 1)(x^2 + px + q) = x^4 + px^3 + qx^2 - x^3 - px^2 - qx + x^2 + px + q
\][/tex]
Combine like terms:
[tex]\[
x^4 + (p-1)x^3 + (q-p+1)x^2 + (p-q)x + q
\][/tex]
5. Equate Coefficients:
Equate the expanded polynomial to [tex]\( x^4 + ax^2 + b \)[/tex] and match coefficients:
[tex]\[
x^4 + (p-1)x^3 + (q-p+1)x^2 + (p-q)x + q = x^4 + ax^2 + b
\][/tex]
This gives us the system of equations:
[tex]\[
\begin{cases}
p - 1 = 0 \\
q - p + 1 = a \\
p - q = 0 \\
q = b
\end{cases}
\][/tex]
6. Solve the System of Equations:
From [tex]\( p - 1 = 0 \)[/tex], we get:
[tex]\[
p = 1
\][/tex]
From [tex]\( p - q = 0 \)[/tex], substituting [tex]\( p = 1 \)[/tex], we get:
[tex]\[
1 - q = 0 \implies q = 1
\][/tex]
Substituting [tex]\( q = 1 \)[/tex] and [tex]\( p = 1 \)[/tex] into [tex]\( q - p + 1 = a \)[/tex]:
[tex]\[
1 - 1 + 1 = a \implies a = 1
\][/tex]
And since [tex]\( q = b \)[/tex]:
[tex]\[
b = 1
\][/tex]
### Final Answer:
The values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that make [tex]\( x^2 - x + 1 \)[/tex] a factor of [tex]\( x^4 + a x^2 + b \)[/tex] are both:
[tex]\[
a = 1, \quad b = 1
\][/tex]
