Answer :
To divide the polynomial [tex]\(\frac{8x^5 + 67x^4 + 160x^3 + 35x^2 - 198x - 72}{8x^2 + 19x + 6}\)[/tex], we follow these steps using polynomial long division:
### Step 1: Set Up the Division
We want to divide the numerator [tex]\(8x^5 + 67x^4 + 160x^3 + 35x^2 - 198x - 72\)[/tex] by the denominator [tex]\(8x^2 + 19x + 6\)[/tex].
### Step 2: Divide the Leading Terms
Look at the leading terms of the numerator and denominator:
- Leading term of the numerator: [tex]\(8x^5\)[/tex]
- Leading term of the denominator: [tex]\(8x^2\)[/tex]
To find the first term of the quotient, divide these leading terms:
[tex]\[
\frac{8x^5}{8x^2} = x^3
\][/tex]
### Step 3: Multiply and Subtract
Multiply the entire denominator by the first term of the quotient ([tex]\(x^3\)[/tex]):
[tex]\[
(8x^2 + 19x + 6) \cdot x^3 = 8x^5 + 19x^4 + 6x^3
\][/tex]
Subtract this from the original numerator:
[tex]\[
(8x^5 + 67x^4 + 160x^3 + 35x^2 - 198x - 72) - (8x^5 + 19x^4 + 6x^3) = 48x^4 + 154x^3 + 35x^2 - 198x - 72
\][/tex]
### Step 4: Repeat the Process
Now, repeat the process with the new polynomial [tex]\(48x^4 + 154x^3 + 35x^2 - 198x - 72\)[/tex].
Divide the leading terms:
[tex]\[
\frac{48x^4}{8x^2} = 6x^2
\][/tex]
Multiply and subtract:
Multiply [tex]\(8x^2 + 19x + 6\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[
(8x^2 + 19x + 6) \cdot 6x^2 = 48x^4 + 114x^3 + 36x^2
\][/tex]
Subtract:
[tex]\[
(48x^4 + 154x^3 + 35x^2 - 198x - 72) - (48x^4 + 114x^3 + 36x^2) = 40x^3 - x^2 - 198x - 72
\][/tex]
### Step 5: Continue Division
Continue with the new polynomial:
[tex]\[
\frac{40x^3}{8x^2} = 5x
\][/tex]
Multiply and subtract:
[tex]\[
(8x^2 + 19x + 6) \cdot 5x = 40x^3 + 95x^2 + 30x
\][/tex]
Subtract:
[tex]\[
(40x^3 - x^2 - 198x - 72) - (40x^3 + 95x^2 + 30x) = -96x^2 - 228x - 72
\][/tex]
### Step 6: Final Steps
Finally, divide the new leading term:
[tex]\[
\frac{-96x^2}{8x^2} = -12
\][/tex]
Multiply and subtract:
[tex]\[
(8x^2 + 19x + 6) \cdot (-12) = -96x^2 - 228x - 72
\][/tex]
Subtract:
[tex]\[
(-96x^2 - 228x - 72) - (-96x^2 - 228x - 72) = 0
\][/tex]
### Conclusion
The division process results in a quotient of [tex]\(x^3 + 6x^2 + 5x - 12\)[/tex] with a remainder of 0. Hence, the expression simplifies to:
[tex]\[
x^3 + 6x^2 + 5x - 12
\][/tex]
### Step 1: Set Up the Division
We want to divide the numerator [tex]\(8x^5 + 67x^4 + 160x^3 + 35x^2 - 198x - 72\)[/tex] by the denominator [tex]\(8x^2 + 19x + 6\)[/tex].
### Step 2: Divide the Leading Terms
Look at the leading terms of the numerator and denominator:
- Leading term of the numerator: [tex]\(8x^5\)[/tex]
- Leading term of the denominator: [tex]\(8x^2\)[/tex]
To find the first term of the quotient, divide these leading terms:
[tex]\[
\frac{8x^5}{8x^2} = x^3
\][/tex]
### Step 3: Multiply and Subtract
Multiply the entire denominator by the first term of the quotient ([tex]\(x^3\)[/tex]):
[tex]\[
(8x^2 + 19x + 6) \cdot x^3 = 8x^5 + 19x^4 + 6x^3
\][/tex]
Subtract this from the original numerator:
[tex]\[
(8x^5 + 67x^4 + 160x^3 + 35x^2 - 198x - 72) - (8x^5 + 19x^4 + 6x^3) = 48x^4 + 154x^3 + 35x^2 - 198x - 72
\][/tex]
### Step 4: Repeat the Process
Now, repeat the process with the new polynomial [tex]\(48x^4 + 154x^3 + 35x^2 - 198x - 72\)[/tex].
Divide the leading terms:
[tex]\[
\frac{48x^4}{8x^2} = 6x^2
\][/tex]
Multiply and subtract:
Multiply [tex]\(8x^2 + 19x + 6\)[/tex] by [tex]\(6x^2\)[/tex]:
[tex]\[
(8x^2 + 19x + 6) \cdot 6x^2 = 48x^4 + 114x^3 + 36x^2
\][/tex]
Subtract:
[tex]\[
(48x^4 + 154x^3 + 35x^2 - 198x - 72) - (48x^4 + 114x^3 + 36x^2) = 40x^3 - x^2 - 198x - 72
\][/tex]
### Step 5: Continue Division
Continue with the new polynomial:
[tex]\[
\frac{40x^3}{8x^2} = 5x
\][/tex]
Multiply and subtract:
[tex]\[
(8x^2 + 19x + 6) \cdot 5x = 40x^3 + 95x^2 + 30x
\][/tex]
Subtract:
[tex]\[
(40x^3 - x^2 - 198x - 72) - (40x^3 + 95x^2 + 30x) = -96x^2 - 228x - 72
\][/tex]
### Step 6: Final Steps
Finally, divide the new leading term:
[tex]\[
\frac{-96x^2}{8x^2} = -12
\][/tex]
Multiply and subtract:
[tex]\[
(8x^2 + 19x + 6) \cdot (-12) = -96x^2 - 228x - 72
\][/tex]
Subtract:
[tex]\[
(-96x^2 - 228x - 72) - (-96x^2 - 228x - 72) = 0
\][/tex]
### Conclusion
The division process results in a quotient of [tex]\(x^3 + 6x^2 + 5x - 12\)[/tex] with a remainder of 0. Hence, the expression simplifies to:
[tex]\[
x^3 + 6x^2 + 5x - 12
\][/tex]
