Verified Answer :
- Decompose the expression \(\frac{15x^{13} - 15x^{11} - 20x^8}{5x^4}\) into three separate quotients: \(\frac{15x^{13}}{5x^4}\), \(-\frac{15x^{11}}{5x^4}\), and \(-\frac{20x^8}{5x^4}\).
- Apply the coefficient division and the quotient rule for exponents (\(x^m/x^n = x^{m-n}\)) to each term.
- Simplify each term to get \(3x^9\), \(-3x^7\), and \(-4x^4\).
- Combine these results to arrive at the final simplified polynomial: \(\boxed{3x^9 - 3x^7 - 4x^4}\).
Explanation
- Decomposing the Expression
The objective is to simplify the rational expression \(\frac{15x^{13} - 15x^{11} - 20x^8}{5x^4}\). This expression involves a polynomial with three terms in the numerator divided by a single monomial in the denominator. To simplify, we can distribute the division by the monomial to each term in the numerator. The rule for dividing a polynomial by a monomial states that \(\frac{A + B + C}{D} = \frac{A}{D} + \frac{B}{D} + \frac{C}{D}\). Applying this here, we have:
\[\frac{15x^{13} - 15x^{11} - 20x^8}{5x^4} = \frac{15x^{13}}{5x^4} - \frac{15x^{11}}{5x^4} - \frac{20x^8}{5x^4}\]
- Simplifying Each Term
Now, we simplify each of the three terms separately by dividing their coefficients and applying the laws of exponents. The law of exponents states that \(\frac{x^m}{x^n} = x^{m-n}\) for \(x eq 0\).
For the first term:
\[\frac{15x^{13}}{5x^4} = \frac{15}{5} \cdot x^{13-4} = 3x^9\]
For the second term:
\[\frac{-15x^{11}}{5x^4} = \frac{-15}{5} \cdot x^{11-4} = -3x^7\]
For the third term:
\[\frac{-20x^8}{5x^4} = \frac{-20}{5} \cdot x^{8-4} = -4x^4\]
- Combining the Simplified Terms
Finally, we combine these individual terms to obtain the simplified expression. Adding these results together, we get:
\[3x^9 - 3x^7 - 4x^4\]
Thus, the expression \(\frac{15x^{13} - 15x^{11} - 20x^8}{5x^4}\) simplifies completely to \(\boxed{3x^9 - 3x^7 - 4x^4}\).
Examples
In computer science and digital signal processing, simplifying algebraic expressions is vital for optimizing algorithms and reducing memory usage. For instance, when a system processes polynomial-based data (like in graphics rendering or data compression), dividing complex polynomial signals by a scaling factor represented by a monomial allows the system to compute results more efficiently. By simplifying expressions like \(\frac{15x^{13} - 15x^{11} - 20x^8}{5x^4}\) into \(3x^9 - 3x^7 - 4x^4\), the computational load is drastically reduced because the system performs three simpler exponentiation operations instead of a more complex division.