Perform the division.
$\frac{10 x^3+35 x^2-5 x+15}{5}= \square$ (Simplify your answer.)
Answer (2)
To divide the polynomial 10 x 3 + 35 x 2 − 5 x + 15 by 5, we divide each term by 5, resulting in 2 x 3 + 7 x 2 − x + 3 . This shows the simplified expression after performing the division. The final result of this operation is 2 x 3 + 7 x 2 − x + 3 .
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Divide each term of the polynomial by 5.
Simplify each term: 5 10 x 3 = 2 x 3 , 5 35 x 2 = 7 x 2 , 5 − 5 x = − x , 5 15 = 3 .
Combine the simplified terms.
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. This means we need to divide each term of the polynomial by 5.
Dividing each term We divide each term of the polynomial by 5: 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
Final result Combining these simplified terms, we get the final result: 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 example, engineers use polynomial division to analyze and design systems, such as control systems. Imagine you're designing a bridge, and you need to calculate the load distribution. Polynomials can represent the forces acting on the bridge, and by dividing these polynomials, you can determine the stress at different points. This ensures the bridge is safe and stable. Similarly, in computer graphics, polynomial division can be used to create smooth curves and surfaces, which are essential for realistic rendering.
