To divide the polynomial \(15x^3 - 17x + 9\) by the polynomial \(3x^2 - 4\), we use polynomial long division. Here is the step-by-step solution:
1. Set up the division:
We are dividing \(15x^3 - 17x + 9\) by \(3x^2 - 4\).
2. Determine the first term of the quotient:
- Divide the leading term of the numerator \(15x^3\) by the leading term of the denominator \(3x^2\):
[tex]\[
\frac{15x^3}{3x^2} = 5x
\][/tex]
- So, the first term of the quotient is \(5x\).
3. Multiply and subtract:
- Multiply the entire denominator \(3x^2 - 4\) by the first term of the quotient \(5x\):
[tex]\[
5x \cdot (3x^2 - 4) = 15x^3 - 20x
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(15x^3 - 17x + 9) - (15x^3 - 20x) = -17x + 20x + 9 = 3x + 9
\][/tex]
4. Result of the division:
- The quotient so far is \(5x\).
- The remainder is \(3x + 9\).
5. Express the final result:
- The original division problem can now be expressed as:
[tex]\[
\frac{15x^3 - 17x + 9}{3x^2 - 4} = 5x + \frac{3x + 9}{3x^2 - 4}
\][/tex]
The final answer is:
[tex]\[
\frac{15x^3 - 17x + 9}{3x^2 - 4} = 5x + \frac{3x + 9}{3x^2 - 4}
\][/tex]
