Perform the division.
$\frac{10 x^3+35 x^2-5 x+15}{5}= \square$ (Simplify your answer.)
Answer (2)
After dividing the polynomial 10 x 3 + 35 x 2 − 5 x + 15 by 5 , we get the simplified result of 2 x 3 + 7 x 2 − x + 3 . This involves dividing each term of the polynomial individually by 5 and then combining the results. The final expression summarizes the outcome of the operation.
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Divide each term of the polynomial by 5.
Simplify each resulting term.
Combine the simplified terms to obtain the final expression.
The result of the division is 2 x 3 + 7 x 2 − x + 3 .
Explanation
Understanding the problem We are asked to perform the division of the polynomial 10 x 3 + 35 x 2 − 5 x + 15 by the constant 5 . The result should be simplified.
Dividing each term To divide the polynomial by 5 , we divide each term of the polynomial by 5 . 5 10 x 3 + 35 x 2 − 5 x + 15 = 5 10 x 3 + 5 35 x 2 − 5 5 x + 5 15
Simplifying the terms Now, we simplify each term: 5 10 x 3 = 2 x 3 5 35 x 2 = 7 x 2 5 − 5 x = − x 5 15 = 3
Combining the terms Combining the simplified terms, we get the final expression: 2 x 3 + 7 x 2 − x + 3
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For instance, engineers use polynomial division to analyze the stability of systems, such as bridges or electrical circuits. By representing the system's behavior as a polynomial, engineers can use division to simplify the analysis and identify potential points of failure or instability. This allows them to design safer and more reliable structures and systems.
