Question

Divide using long division. State the​ quotient, q(x), and the​ remainder, r(x).
​(15 x squared minus 13 x minus 13​)divided by​(3 x minus 5​)
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Part 1
​(15 x squared minus 13 x minus 13​)divided by​(3 x minus 5​)equals
  
enter your response hereplusStartFraction nothing Over 3 x minus 5 EndFraction
  
​(Simplify your answers. Do not​ factor.)

Answer 1

To perform long division on \(15x^2 - 13x - 13\) divided by \(3x - 5\), follow these steps:

1. **Set up the long division**: Write \(15x^2 - 13x - 13\) under the long division symbol and \(3x - 5\) outside.

2. **Divide the first term**: Divide the leading term of the dividend \(15x^2\) by the leading term of the divisor \(3x\):
\[
\frac{15x^2}{3x} = 5x
\]
This will be the first term of the quotient \(q(x)\).

3. **Multiply**: Multiply \(5x\) by the entire divisor \(3x - 5\):
\[
5x(3x - 5) = 15x^2 - 25x
\]

4. **Subtract**: Now subtract this product from the original polynomial:
\[
(15x^2 - 13x - 13) - (15x^2 - 25x) = (15x^2 - 15x^2) + (-13x + 25x) - 13 = 12x - 13
\]

5. **Repeat the process**: Now repeat the process with \(12x - 13\):
- Divide \(12x\) by \(3x\):
\[
\frac{12x}{3x} = 4
\]
Add this to the quotient \(q(x)\). So now, \(q(x) = 5x + 4\).

6. **Multiply**: Multiply \(4\) by \(3x - 5\):
\[
4(3x - 5) = 12x - 20
\]

7. **Subtract again**:
\[
(12x - 13) - (12x - 20) = (12x - 12x) + (-13 + 20) = 7
\]

8. **Final result**: The division process stops here because the degree of the remainder \(7\) is less than the degree of the divisor \(3x - 5\).

The quotient \(q(x)\) and the remainder \(r(x)\) are:
\[
q(x) = 5x + 4
\]
\[
r(x) = 7
\]

Putting it together, the final answer is:
\[
(15x^2 - 13x - 13) \div (3x - 5) = 5x + 4 + \frac{7}{3x - 5}
\]