Answer :
To solve the polynomial division problem [tex]\(\frac{6x^4 - 19x^3 - 56x^2 + 35}{6x + 5}\)[/tex], we perform polynomial long division to find the quotient and remainder.
#### Step 1: Set up the division
The dividend is [tex]\(6x^4 - 19x^3 - 56x^2 + 0x + 35\)[/tex], and the divisor is [tex]\(6x + 5\)[/tex].
#### Step 2: Divide the first term
- Divide the leading term of the dividend ([tex]\(6x^4\)[/tex]) by the leading term of the divisor ([tex]\(6x\)[/tex]), which gives you [tex]\(x^3\)[/tex].
- Multiply the entire divisor by this result: [tex]\((6x + 5) \times x^3 = 6x^4 + 5x^3\)[/tex].
#### Step 3: Subtract
- Subtract [tex]\(6x^4 + 5x^3\)[/tex] from the original dividend:
[tex]\((6x^4 - 19x^3) - (6x^4 + 5x^3) = -24x^3\)[/tex].
#### Step 4: Bring down the next term
- Bring down the [tex]\(-56x^2\)[/tex] to get: [tex]\(-24x^3 - 56x^2\)[/tex].
#### Step 5: Repeat
Continue this process:
1. Divide [tex]\(-24x^3\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(-4x^2\)[/tex].
2. Multiply: [tex]\((6x + 5) \times -4x^2 = -24x^3 - 20x^2\)[/tex].
3. Subtract to get: [tex]\(-56x^2 - (-20x^2) = -36x^2\)[/tex].
4. Bring down the next term, [tex]\(0x\)[/tex], to get: [tex]\(-36x^2 + 0x\)[/tex].
5. Divide [tex]\(-36x^2\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(-6x\)[/tex].
6. Multiply: [tex]\((6x + 5) \times -6x = -36x^2 - 30x\)[/tex].
7. Subtract to get: [tex]\(0x - (-30x) = 30x\)[/tex].
8. Bring down the last constant term, [tex]\(35\)[/tex], to get: [tex]\(30x + 35\)[/tex].
9. Divide [tex]\(30x\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(5\)[/tex].
10. Multiply: [tex]\((6x + 5) \times 5 = 30x + 25\)[/tex].
11. Subtract to yield the remainder: [tex]\(35 - 25 = 10\)[/tex].
The quotient of the division is [tex]\(x^3 - 4x^2 - 6x + 5\)[/tex], and the remainder is [tex]\(10\)[/tex].
Thus, the complete division result is:
[tex]\[
\frac{6x^4 - 19x^3 - 56x^2 + 35}{6x + 5} = x^3 - 4x^2 - 6x + 5 \text{ with a remainder of } 10
\][/tex]
This means:
[tex]\[
6x^4 - 19x^3 - 56x^2 + 35 = (6x + 5) \cdot (x^3 - 4x^2 - 6x + 5) + 10
\][/tex]
#### Step 1: Set up the division
The dividend is [tex]\(6x^4 - 19x^3 - 56x^2 + 0x + 35\)[/tex], and the divisor is [tex]\(6x + 5\)[/tex].
#### Step 2: Divide the first term
- Divide the leading term of the dividend ([tex]\(6x^4\)[/tex]) by the leading term of the divisor ([tex]\(6x\)[/tex]), which gives you [tex]\(x^3\)[/tex].
- Multiply the entire divisor by this result: [tex]\((6x + 5) \times x^3 = 6x^4 + 5x^3\)[/tex].
#### Step 3: Subtract
- Subtract [tex]\(6x^4 + 5x^3\)[/tex] from the original dividend:
[tex]\((6x^4 - 19x^3) - (6x^4 + 5x^3) = -24x^3\)[/tex].
#### Step 4: Bring down the next term
- Bring down the [tex]\(-56x^2\)[/tex] to get: [tex]\(-24x^3 - 56x^2\)[/tex].
#### Step 5: Repeat
Continue this process:
1. Divide [tex]\(-24x^3\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(-4x^2\)[/tex].
2. Multiply: [tex]\((6x + 5) \times -4x^2 = -24x^3 - 20x^2\)[/tex].
3. Subtract to get: [tex]\(-56x^2 - (-20x^2) = -36x^2\)[/tex].
4. Bring down the next term, [tex]\(0x\)[/tex], to get: [tex]\(-36x^2 + 0x\)[/tex].
5. Divide [tex]\(-36x^2\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(-6x\)[/tex].
6. Multiply: [tex]\((6x + 5) \times -6x = -36x^2 - 30x\)[/tex].
7. Subtract to get: [tex]\(0x - (-30x) = 30x\)[/tex].
8. Bring down the last constant term, [tex]\(35\)[/tex], to get: [tex]\(30x + 35\)[/tex].
9. Divide [tex]\(30x\)[/tex] by [tex]\(6x\)[/tex] to get [tex]\(5\)[/tex].
10. Multiply: [tex]\((6x + 5) \times 5 = 30x + 25\)[/tex].
11. Subtract to yield the remainder: [tex]\(35 - 25 = 10\)[/tex].
The quotient of the division is [tex]\(x^3 - 4x^2 - 6x + 5\)[/tex], and the remainder is [tex]\(10\)[/tex].
Thus, the complete division result is:
[tex]\[
\frac{6x^4 - 19x^3 - 56x^2 + 35}{6x + 5} = x^3 - 4x^2 - 6x + 5 \text{ with a remainder of } 10
\][/tex]
This means:
[tex]\[
6x^4 - 19x^3 - 56x^2 + 35 = (6x + 5) \cdot (x^3 - 4x^2 - 6x + 5) + 10
\][/tex]
