Perform the division. [tex]$\frac{10 x^5+7 x^3}{x}$[/tex]
Answer (2)
To divide the polynomial 10 x 5 + 7 x 3 by x , we simplify each term separately, resulting in the expression 10 x 4 + 7 x 2 . This involves using exponent rules to reduce the powers. The final answer is 10 x 4 + 7 x 2 .
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Divide each term in the numerator by x : x 10 x 5 + x 7 x 3 .
Simplify each term using exponent rules: 10 x 5 − 1 + 7 x 3 − 1 .
Simplify the exponents: 10 x 4 + 7 x 2 .
The simplified expression is 10 x 4 + 7 x 2 .
Explanation
Understanding the Problem We are asked to perform the division of the polynomial 10 x 5 + 7 x 3 by x . This involves dividing each term of the polynomial by x .
Separating the Terms We can rewrite the expression as: x 10 x 5 + 7 x 3 = x 10 x 5 + x 7 x 3
Simplifying Each Term Now, we simplify each term by using the rule of exponents, which states that x b x a = x a − b . So we have: x 10 x 5 = 10 x 5 − 1 = 10 x 4 x 7 x 3 = 7 x 3 − 1 = 7 x 2
Combining the Results Combining these simplified terms, we get: 10 x 4 + 7 x 2
Final Answer Therefore, the simplified expression is 10 x 4 + 7 x 2 .
Examples
Understanding polynomial division is crucial in various fields, such as engineering and computer science. For instance, when designing filters for signal processing, engineers often use polynomial division to simplify complex transfer functions. Similarly, in computer graphics, polynomial division can be used to optimize rendering algorithms, making them more efficient and faster. This skill also helps in modeling real-world phenomena with greater accuracy.
