To find the quotient and remainder when the polynomial [tex]\(2x^2 - 5x + 3\)[/tex] is divided by [tex]\(x\)[/tex], we can use polynomial long division. Let's go through the steps together:
1. Set up the division. Write the dividend [tex]\(2x^2 - 5x + 3\)[/tex] under the long division symbol, and the divisor [tex]\(x\)[/tex] outside:
[tex]\[
\begin{array}{r|lr}
& 2x^2 - 5x + 3 \\
x &
\end{array}
\][/tex]
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
This gives us the first term of the quotient, [tex]\(2x\)[/tex].
3. Multiply the entire divisor [tex]\(x\)[/tex] by the first term of the quotient [tex]\((2x)\)[/tex] and write the result below the corresponding terms of the dividend:
[tex]\[
2x \cdot x = 2x^2
\][/tex]
[tex]\[
\begin{array}{r|lr}
& 2x^2 - 5x + 3 \\
x & 2x^2
\end{array}
\][/tex]
4. Subtract this result from the original dividend:
[tex]\[
(2x^2 - 5x + 3) - (2x^2) = -5x + 3
\][/tex]
So now, we write the intermediate result:
[tex]\[
\begin{array}{r|lr}
& 2x^2 - 5x + 3 \\
x & 2x^2 \\
& \underline{- 2x^2} \\
& - 5x + 3 \\
\end{array}
\][/tex]
5. Divide the first term of the new intermediate result by the first term of the divisor:
[tex]\[
\frac{-5x}{x} = -5
\][/tex]
This gives us the next term of the quotient, [tex]\(-5\)[/tex].
6. Multiply the entire divisor [tex]\(x\)[/tex] by the next term of the quotient [tex]\((-5)\)[/tex] and write the result below the corresponding terms of the intermediate result:
[tex]\[
-5 \cdot x = -5x
\][/tex]
[tex]\[
\begin{array}{r|lr}
& 2x^2 - 5x + 3 \\
x & 2x^2 - 5 \\
& \underline{- 2x^2} \\
& - 5x + 3 \\
& \underline{- 5x} \\
\end{array}
\][/tex]
7. Subtract this result from the intermediate result:
[tex]\[
(-5x + 3) - (-5x) = 3
\][/tex]
This gives the new intermediate result:
[tex]\[
\begin{array}{r|lr}
& 2x^2 - 5x + 3 \\
x & 2x^2 - 5 \\
& \underline{- 2x^2} \\
& - 5x + 3 \\
& \underline{- 5x} \\
& 3 \\
\end{array}
\][/tex]
8. Since we cannot divide 3 by [tex]\(x\)[/tex] (as 3 is a constant and [tex]\(x\)[/tex] is a variable), 3 becomes the remainder.
Therefore, the quotient is [tex]\(2x - 5\)[/tex] and the remainder is [tex]\(3\)[/tex]. So, our final answer is:
[tex]\[
\boxed{(2x - 5, \text{ remainder } 3)}
\][/tex]
