Answer :
To solve the problem of dividing the polynomial [tex]\((3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9)\)[/tex] by [tex]\((3x - 4)\)[/tex], we perform polynomial long division. Here’s a step-by-step explanation of how you can carry out this division:
### Step 1: Set up the division
The polynomial division is set up similarly to numerical long division. The dividend is [tex]\(3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex] and the divisor is [tex]\(3x - 4\)[/tex].
### Step 2: Divide the leading terms
- Divide the leading term of the dividend, [tex]\(3x^5\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]. This gives you [tex]\(x^4\)[/tex].
- This [tex]\(x^4\)[/tex] is the first term of the quotient.
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(3x - 4\)[/tex] by [tex]\(x^4\)[/tex] to get [tex]\(3x^5 - 4x^4\)[/tex].
- Subtract [tex]\(3x^5 - 4x^4\)[/tex] from the original dividend to create a new polynomial: [tex]\((-31x^4) - (-4x^4)\)[/tex], which simplifies to [tex]\(-27x^4\)[/tex].
- Bring down the next term from the dividend to continue.
### Step 4: Repeat the process
- Divide [tex]\(-27x^4\)[/tex] (the new leading term) by [tex]\(3x\)[/tex] to get [tex]\(-9x^3\)[/tex].
- Multiply the divisor [tex]\(3x - 4\)[/tex] by [tex]\(-9x^3\)[/tex] to get [tex]\(-27x^4 + 36x^3\)[/tex].
- Subtract this result from the polynomial you formed in Step 3.
### Step 5: Continue until done
- Continue in this fashion dividing, multiplying, and subtracting, while bringing down the next term as needed, until all terms have been brought down and the division is complete.
### Final polynomial
After following through all similar steps, you end up with a quotient polynomial and possibly a remainder. Upon completing the steps of division:
Quotient: [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex]
Remainder: [tex]\(-7\)[/tex]
### Conclusion
Thus, dividing [tex]\((3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9)\)[/tex] by [tex]\((3x - 4)\)[/tex] results in a quotient of [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex] with a remainder of [tex]\(-7\)[/tex].
Matching this with the given choices, the correct answer is:
B) [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4, R -7\)[/tex]
### Step 1: Set up the division
The polynomial division is set up similarly to numerical long division. The dividend is [tex]\(3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9\)[/tex] and the divisor is [tex]\(3x - 4\)[/tex].
### Step 2: Divide the leading terms
- Divide the leading term of the dividend, [tex]\(3x^5\)[/tex], by the leading term of the divisor, [tex]\(3x\)[/tex]. This gives you [tex]\(x^4\)[/tex].
- This [tex]\(x^4\)[/tex] is the first term of the quotient.
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(3x - 4\)[/tex] by [tex]\(x^4\)[/tex] to get [tex]\(3x^5 - 4x^4\)[/tex].
- Subtract [tex]\(3x^5 - 4x^4\)[/tex] from the original dividend to create a new polynomial: [tex]\((-31x^4) - (-4x^4)\)[/tex], which simplifies to [tex]\(-27x^4\)[/tex].
- Bring down the next term from the dividend to continue.
### Step 4: Repeat the process
- Divide [tex]\(-27x^4\)[/tex] (the new leading term) by [tex]\(3x\)[/tex] to get [tex]\(-9x^3\)[/tex].
- Multiply the divisor [tex]\(3x - 4\)[/tex] by [tex]\(-9x^3\)[/tex] to get [tex]\(-27x^4 + 36x^3\)[/tex].
- Subtract this result from the polynomial you formed in Step 3.
### Step 5: Continue until done
- Continue in this fashion dividing, multiplying, and subtracting, while bringing down the next term as needed, until all terms have been brought down and the division is complete.
### Final polynomial
After following through all similar steps, you end up with a quotient polynomial and possibly a remainder. Upon completing the steps of division:
Quotient: [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex]
Remainder: [tex]\(-7\)[/tex]
### Conclusion
Thus, dividing [tex]\((3x^5 - 31x^4 + 54x^3 - 30x^2 - 4x + 9)\)[/tex] by [tex]\((3x - 4)\)[/tex] results in a quotient of [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4\)[/tex] with a remainder of [tex]\(-7\)[/tex].
Matching this with the given choices, the correct answer is:
B) [tex]\(x^4 - 9x^3 + 6x^2 - 2x - 4, R -7\)[/tex]
