Answer :
Sure! Let's go through the process of dividing the polynomial [tex]\(3x^4 + 18x^3 + 45x^2 + 60x + 31\)[/tex] by [tex]\(3x + 6\)[/tex] using polynomial long division.
### Step 1: Setup
Place the divisor [tex]\(3x + 6\)[/tex] outside the division bracket and the dividend [tex]\(3x^4 + 18x^3 + 45x^2 + 60x + 31\)[/tex] inside the division bracket.
### Step 2: Divide
Divide the first term of the dividend [tex]\(3x^4\)[/tex] by the first term of the divisor [tex]\(3x\)[/tex].
[tex]\[
\frac{3x^4}{3x} = x^3
\][/tex]
This gives us the first term of the quotient, which is [tex]\(x^3\)[/tex].
### Step 3: Multiply
Multiply the entire divisor [tex]\(3x + 6\)[/tex] by this first term of the quotient [tex]\(x^3\)[/tex], and write the result under the dividend:
[tex]\[
(x^3) \cdot (3x + 6) = 3x^4 + 6x^3
\][/tex]
### Step 4: Subtract
Subtract this result from the dividend:
[tex]\[
(3x^4 + 18x^3 + 45x^2 + 60x + 31) - (3x^4 + 6x^3) = 12x^3 + 45x^2 + 60x + 31
\][/tex]
### Step 5: Repeat the Process
Now, divide the new dividend [tex]\(12x^3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{12x^3}{3x} = 4x^2
\][/tex]
This gives the next term of the quotient: [tex]\(4x^2\)[/tex].
Multiply [tex]\(3x + 6\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
(4x^2) \cdot (3x + 6) = 12x^3 + 24x^2
\][/tex]
Subtract the result:
[tex]\[
(12x^3 + 45x^2 + 60x + 31) - (12x^3 + 24x^2) = 21x^2 + 60x + 31
\][/tex]
### Step 6: Continue Division
Divide [tex]\(21x^2\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{21x^2}{3x} = 7x
\][/tex]
Add [tex]\(7x\)[/tex] to the quotient, and multiply:
[tex]\[
(7x) \cdot (3x + 6) = 21x^2 + 42x
\][/tex]
Subtract again:
[tex]\[
(21x^2 + 60x + 31) - (21x^2 + 42x) = 18x + 31
\][/tex]
### Step 7: Final Step
Divide [tex]\(18x\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{18x}{3x} = 6
\][/tex]
Add 6 to the quotient, and multiply:
[tex]\[
6 \cdot (3x + 6) = 18x + 36
\][/tex]
Subtract the final time:
[tex]\[
(18x + 31) - (18x + 36) = -5
\][/tex]
### Conclusion
The quotient is [tex]\(x^3 + 4x^2 + 7x + 6\)[/tex], and the remainder is [tex]\(-5\)[/tex].
Thus, the division of [tex]\(3x^4 + 18x^3 + 45x^2 + 60x + 31\)[/tex] by [tex]\(3x + 6\)[/tex] gives a quotient of [tex]\(x^3 + 4x^2 + 7x + 6\)[/tex] with a remainder of [tex]\(-5\)[/tex], or:
[tex]\[
(3x^4 + 18x^3 + 45x^2 + 60x + 31) = (3x + 6)(x^3 + 4x^2 + 7x + 6) - 5
\][/tex]
### Step 1: Setup
Place the divisor [tex]\(3x + 6\)[/tex] outside the division bracket and the dividend [tex]\(3x^4 + 18x^3 + 45x^2 + 60x + 31\)[/tex] inside the division bracket.
### Step 2: Divide
Divide the first term of the dividend [tex]\(3x^4\)[/tex] by the first term of the divisor [tex]\(3x\)[/tex].
[tex]\[
\frac{3x^4}{3x} = x^3
\][/tex]
This gives us the first term of the quotient, which is [tex]\(x^3\)[/tex].
### Step 3: Multiply
Multiply the entire divisor [tex]\(3x + 6\)[/tex] by this first term of the quotient [tex]\(x^3\)[/tex], and write the result under the dividend:
[tex]\[
(x^3) \cdot (3x + 6) = 3x^4 + 6x^3
\][/tex]
### Step 4: Subtract
Subtract this result from the dividend:
[tex]\[
(3x^4 + 18x^3 + 45x^2 + 60x + 31) - (3x^4 + 6x^3) = 12x^3 + 45x^2 + 60x + 31
\][/tex]
### Step 5: Repeat the Process
Now, divide the new dividend [tex]\(12x^3\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{12x^3}{3x} = 4x^2
\][/tex]
This gives the next term of the quotient: [tex]\(4x^2\)[/tex].
Multiply [tex]\(3x + 6\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
(4x^2) \cdot (3x + 6) = 12x^3 + 24x^2
\][/tex]
Subtract the result:
[tex]\[
(12x^3 + 45x^2 + 60x + 31) - (12x^3 + 24x^2) = 21x^2 + 60x + 31
\][/tex]
### Step 6: Continue Division
Divide [tex]\(21x^2\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{21x^2}{3x} = 7x
\][/tex]
Add [tex]\(7x\)[/tex] to the quotient, and multiply:
[tex]\[
(7x) \cdot (3x + 6) = 21x^2 + 42x
\][/tex]
Subtract again:
[tex]\[
(21x^2 + 60x + 31) - (21x^2 + 42x) = 18x + 31
\][/tex]
### Step 7: Final Step
Divide [tex]\(18x\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
\frac{18x}{3x} = 6
\][/tex]
Add 6 to the quotient, and multiply:
[tex]\[
6 \cdot (3x + 6) = 18x + 36
\][/tex]
Subtract the final time:
[tex]\[
(18x + 31) - (18x + 36) = -5
\][/tex]
### Conclusion
The quotient is [tex]\(x^3 + 4x^2 + 7x + 6\)[/tex], and the remainder is [tex]\(-5\)[/tex].
Thus, the division of [tex]\(3x^4 + 18x^3 + 45x^2 + 60x + 31\)[/tex] by [tex]\(3x + 6\)[/tex] gives a quotient of [tex]\(x^3 + 4x^2 + 7x + 6\)[/tex] with a remainder of [tex]\(-5\)[/tex], or:
[tex]\[
(3x^4 + 18x^3 + 45x^2 + 60x + 31) = (3x + 6)(x^3 + 4x^2 + 7x + 6) - 5
\][/tex]
