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Divide:

[tex]\[ \left(15x^3 - 17x + 9\right) \div \left(3x^2 - 4\right) \][/tex]

Write your answer in the following form: Quotient [tex]\[+\frac{\text{Remainder}}{3x^2 - 4}\][/tex].

[tex]\[ \frac{15x^3 - 17x + 9}{3x^2 - 4} = \square + \frac{\square}{3x^2 - 4} \][/tex]



Answer :

GinnyAnswer
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]

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