To find the quotient and remainder of the division of [tex]\( \frac{2x^3 - 14x^2 + 7x - 33}{2x^2 + 5} \)[/tex] using polynomial long division, we will go through the process step-by-step.
### Step-by-Step Solution:
1. Set up the division:
We need to divide the polynomial [tex]\( 2x^3 - 14x^2 + 7x - 33 \)[/tex] by [tex]\( 2x^2 + 5 \)[/tex].
2. First Division:
- Divide the leading term of the numerator ([tex]\(2x^3\)[/tex]) by the leading term of the denominator ([tex]\(2x^2\)[/tex]):
[tex]\[
\frac{2x^3}{2x^2} = x
\][/tex]
- The first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire denominator [tex]\( 2x^2 + 5 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[
x \cdot (2x^2 + 5) = 2x^3 + 5x
\][/tex]
- Subtract this result from the original numerator:
[tex]\[
(2x^3 - 14x^2 + 7x - 33) - (2x^3 + 5x) = -14x^2 + 2x - 33
\][/tex]
4. Second Division:
- Divide the new leading term of the numerator ([tex]\(-14x^2\)[/tex]) by the leading term of the denominator ([tex]\(2x^2\)[/tex]):
[tex]\[
\frac{-14x^2}{2x^2} = -7
\][/tex]
- The next term of the quotient is [tex]\(-7\)[/tex].
5. Multiply and Subtract Again:
- Multiply the entire denominator [tex]\( 2x^2 + 5 \)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[
-7 \cdot (2x^2 + 5) = -14x^2 - 35
\][/tex]
- Subtract this from the current numerator:
[tex]\[
(-14x^2 + 2x - 33) - (-14x^2 - 35) = 2x - 33 + 35 = 2x + 2
\][/tex]
6. Conclusion:
- The quotient is the sum of the terms we obtained during division: [tex]\( x - 7 \)[/tex].
- The remainder is what is left after the last subtraction: [tex]\( 2x + 2 \)[/tex].
Thus, the final results are:
- The quotient is [tex]\( x - 7 \)[/tex].
- The remainder is [tex]\( 2x + 2 \)[/tex].
