Sure! Let's go through the process of dividing the polynomial [tex]\(\frac{2x^3 + x^2 - 3}{2x - 1}\)[/tex] using long division. Here's a step-by-step solution:
### Step 1: Set up the division
We set up the long division by writing [tex]\(2x - 1\)[/tex] (the divisor) to the left and [tex]\(2x^3 + x^2 - 3\)[/tex] (the dividend) under the division symbol.
### Step 2: Divide the leading terms
To find the first term of the quotient, divide the leading term of the dividend, [tex]\(2x^3\)[/tex], by the leading term of the divisor, [tex]\(2x\)[/tex]. This gives us:
[tex]\[\frac{2x^3}{2x} = x^2\][/tex]
Write [tex]\(x^2\)[/tex] above the division bar.
### Step 3: Multiply and subtract
Multiply [tex]\(x^2\)[/tex] by [tex]\(2x - 1\)[/tex] and subtract from the dividend:
[tex]\[
(x^2) \cdot (2x - 1) = 2x^3 - x^2
\][/tex]
Now subtract this from the dividend:
[tex]\[
(2x^3 + x^2 - 3) - (2x^3 - x^2) = 2x^2 - 3
\][/tex]
### Step 4: Repeat the process
Now, we repeat the process with the new dividend [tex]\(2x^2 - 3\)[/tex]:
Divide the leading term [tex]\(2x^2\)[/tex] by the leading term [tex]\(2x\)[/tex]:
[tex]\[
\frac{2x^2}{2x} = x
\][/tex]
Write [tex]\(x\)[/tex] above the division bar next to [tex]\(x^2\)[/tex].
Multiply [tex]\(x\)[/tex] by [tex]\(2x - 1\)[/tex]:
[tex]\[
(x) \cdot (2x - 1) = 2x^2 - x
\][/tex]
Subtract this from the current dividend:
[tex]\[
(2x^2 - 3) - (2x^2 - x) = x - 3
\][/tex]
### Step 5: Repeat the process again
Now, our new dividend is [tex]\(x - 3\)[/tex]:
Divide the leading term [tex]\(x\)[/tex] by the leading term [tex]\(2x\)[/tex]:
[tex]\[
\frac{x}{2x} = \frac{1}{2}
\][/tex]
Write [tex]\(\frac{1}{2}\)[/tex] above the division bar next to [tex]\(x\)[/tex].
Multiply [tex]\(\frac{1}{2}\)[/tex] by [tex]\(2x - 1\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right) \cdot (2x - 1) = x - \frac{1}{2}
\][/tex]
Subtract this from the current dividend:
[tex]\[
(x - 3) - \left(x - \frac{1}{2}\right) = -\frac{5}{2}
\][/tex]
### Summary
After putting it all together, our quotient is:
[tex]\[ x^2 + x + \frac{1}{2} \][/tex]
And our remainder is:
[tex]\[ -\frac{5}{2} \][/tex]
So, the division of [tex]\(\frac{2x^3 + x^2 - 3}{2x - 1}\)[/tex] gives:
[tex]\[ \frac{2x^3 + x^2 - 3}{2x - 1} = x^2 + x + \frac{1}{2} + \frac{-\frac{5}{2}}{2x - 1} \][/tex]
The quotient is [tex]\(x^2 + x + \frac{1}{2}\)[/tex] and the remainder is [tex]\(-\frac{5}{2}\)[/tex].
