To divide [tex]\( p(x) = x^4 - 3x^2 + 4u + 5 \)[/tex] by [tex]\( g(x) = x^2 + 1 - x \)[/tex], we'll perform polynomial long division step-by-step.
### Step 1: Arrange Polynomials
First, we ensure both polynomials are in standard form with descending powers of [tex]\( x \)[/tex]:
[tex]\( p(x) = x^4 + 0x^3 - 3x^2 + 0x + (4u + 5) \)[/tex]
[tex]\( g(x) = x^2 - x + 1 \)[/tex]
### Step 2: Divide the Leading Terms
Divide the leading term of [tex]\( p(x) \)[/tex] by the leading term of [tex]\( g(x) \)[/tex]:
[tex]\[
\frac{x^4}{x^2} = x^2
\][/tex]
This is the first term of the quotient.
### Step 3: Multiply and Subtract
Multiply [tex]\( g(x) \)[/tex] by [tex]\( x^2 \)[/tex] and subtract from [tex]\( p(x) \)[/tex]:
[tex]\[
x^2 \cdot (x^2 - x + 1) = x^4 - x^3 + x^2
\][/tex]
[tex]\[
p(x) - (x^4 - x^3 + x^2) = (x^4 - 3x^2 + 4u + 5) - (x^4 - x^3 + x^2) = x^3 - 4x^2 + 4u + 5
\][/tex]
### Step 4: Repeat the Process
Now, consider the new polynomial [tex]\( x^3 - 4x^2 + 4u + 5 \)[/tex] and divide the leading term by the leading term of [tex]\( g(x) \)[/tex]:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
This is the next term of the quotient.
### Step 5: Multiply and Subtract Again
Multiply [tex]\( g(x) \)[/tex] by [tex]\( x \)[/tex] and subtract from the current polynomial:
[tex]\[
x \cdot (x^2 - x + 1) = x^3 - x^2 + x
\][/tex]
[tex]\[
(x^3 - 4x^2 + 4u + 5) - (x^3 - x^2 + x) = -3x^2 - x + 4u + 5
\][/tex]
### Step 6: Continue the Process
Continue the division process:
[tex]\[
\frac{-3x^2}{x^2} = -3
\][/tex]
This is the next term of the quotient.
Multiply [tex]\( g(x) \)[/tex] by [tex]\(-3\)[/tex] and subtract:
[tex]\[
-3 \cdot (x^2 - x + 1) = -3x^2 + 3x - 3
\][/tex]
[tex]\[
(-3x^2 - x + 4u + 5) - (-3x^2 + 3x - 3) = -x - 4x + 4u + 8
\][/tex]
[tex]\(
-4x + (4u + 8)
\)[/tex]
Finally, the quotient is:
[tex]\[
x^2 + x - 3
\][/tex]
And the remainder is:
[tex]\[
4u - 4x + 8
\][/tex]
### Conclusion
The quotient when [tex]\( p(x) \)[/tex] is divided by [tex]\( g(x) \)[/tex] is:
[tex]\[
x^2 + x - 3
\][/tex]
And the remainder is:
[tex]\[
4u - 4x + 8
\][/tex]
