To find the quotient and remainder of the division [tex]\(\frac{-30x^2 + 60x - 38}{5x - 5}\)[/tex], we will perform polynomial long division.
### Step-by-Step Long Division Process
1. Setup the Division:
Divide the polynomial [tex]\(-30x^2 + 60x - 38\)[/tex] by the polynomial [tex]\(5x - 5\)[/tex].
2. First Term of the Quotient:
Divide the leading term of the numerator [tex]\(-30x^2\)[/tex] by the leading term of the denominator [tex]\(5x\)[/tex]:
[tex]\[
\frac{-30x^2}{5x} = -6x
\][/tex]
So, the first term of the quotient is [tex]\(-6x\)[/tex].
3. Multiply and Subtract:
Multiply [tex]\(-6x\)[/tex] by [tex]\(5x - 5\)[/tex]:
[tex]\[
(-6x) \cdot (5x - 5) = -30x^2 + 30x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(-30x^2 + 60x - 38) - (-30x^2 + 30x) = 30x - 38
\][/tex]
4. Second Term of the Quotient:
Divide the new leading term [tex]\(30x\)[/tex] by the leading term [tex]\(5x\)[/tex]:
[tex]\[
\frac{30x}{5x} = 6
\][/tex]
So, the next term of the quotient is [tex]\(6\)[/tex].
5. Multiply and Subtract:
Multiply [tex]\(6\)[/tex] by [tex]\(5x - 5\)[/tex]:
[tex]\[
(6) \cdot (5x - 5) = 30x - 30
\][/tex]
Subtract this from the current polynomial:
[tex]\[
(30x - 38) - (30x - 30) = -8
\][/tex]
### Conclusion:
No further terms in the numerator are left to divide. Therefore, the process ends here.
- Quotient: The quotient is obtained from the terms we calculated during the process, so it is [tex]\(-6x + 6\)[/tex], which can be simplified as [tex]\([-6.0, 6.0]\)[/tex].
- Remainder: The remainder, which is left after the last subtraction, is [tex]\(-8\)[/tex], which can be written as [tex]\([-8.0]\)[/tex].
Thus, the final results are:
- The quotient is [tex]\((-6x + 6)\)[/tex], which corresponds to [tex]\([-6.0, 6.0]\)[/tex].
- The remainder is [tex]\(-8\)[/tex], which corresponds to [tex]\([-8.0]\)[/tex].
So, the quotient is [tex]\([-6.0, 6.0]\)[/tex] and the remainder is [tex]\([-8.0]\)[/tex].
