To find the remainder when dividing the polynomial \(4x^4 - 6x^3 + 6x^2 - 1\) by the polynomial \(2x^2 - 3\), let's go through the process of polynomial division step-by-step.
### Step 1: Set Up the Division
We start with the dividend \(4x^4 - 6x^3 + 6x^2 - 1\) and the divisor \(2x^2 - 3\).
### Step 2: Divide the Leading Terms
First, we divide the leading term of the dividend, \(4x^4\), by the leading term of the divisor, \(2x^2\):
[tex]\[
\frac{4x^4}{2x^2} = 2x^2
\][/tex]
This gives us the first term of the quotient, which is \(2x^2\).
### Step 3: Multiply and Subtract
Next, we multiply the entire divisor by this term of the quotient:
[tex]\[
(2x^2)(2x^2 - 3) = 4x^4 - 6x^2
\][/tex]
Now, subtract this product from the current dividend:
[tex]\[
(4x^4 - 6x^3 + 6x^2 - 1) - (4x^4 - 6x^2) = -6x^3 + 12x^2 - 1
\][/tex]
### Step 4: Repeat the Process
Now, we repeat the process with the new polynomial \(-6x^3 + 12x^2 - 1\).
#### Divide the Leading Terms
[tex]\[
\frac{-6x^3}{2x^2} = -3x
\][/tex]
This gives us the next term of the quotient, which is \(-3x\).
#### Multiply and Subtract
[tex]\[
(-3x)(2x^2 - 3) = -6x^3 + 9x
\][/tex]
Now, subtract this product from the current polynomial:
[tex]\[
(-6x^3 + 12x^2 - 1) - (-6x^3 + 9x) = 12x^2 - 9x - 1
\][/tex]
### Step 5: Continue the Process
We continue with the polynomial \(12x^2 - 9x - 1\).
#### Divide the Leading Terms
[tex]\[
\frac{12x^2}{2x^2} = 6
\][/tex]
This gives us the next term of the quotient, which is \(6\).
#### Multiply and Subtract
[tex]\[
(6)(2x^2 - 3) = 12x^2 - 18
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
(12x^2 - 9x - 1) - (12x^2 - 18) = -9x + 17
\][/tex]
### Conclusion
Now, we cannot divide further as the degree of the remainder \(-9x + 17\) is less than the degree of the divisor \(2x^2 - 3\). Therefore, the quotient is \(2x^2 - 3x + 6\), and the remainder is \(17 -9 x\).
The final result of the division is:
[tex]\[
\boxed{(2x^2 - 3x + 6, 17 - 9x)}
\][/tex]
where [tex]\(2x^2 - 3x + 6\)[/tex] is the quotient and [tex]\(17 - 9x\)[/tex] is the remainder.
