Divide.

$\left(2 x^3+2 x-15\right) \text { by }(x+5)$

The answer is $\square$ .

The result of dividing 2 x 3 + 2 x − 15 by x + 5 is 2 x 2 − 10 x + 52 − x + 5 275 ​ . This includes a polynomial quotient and a remainder. The long division method helps break down the process step-by-step.
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Answered by Anonymous | 2025-07-04

Perform polynomial long division of ( 2 x 3 + 2 x − 15 ) by ( x + 5 ) .
Divide 2 x 3 by x to get 2 x 2 , multiply ( x + 5 ) by 2 x 2 to get 2 x 3 + 10 x 2 , and subtract from the dividend to get − 10 x 2 + 2 x − 15 .
Divide − 10 x 2 by x to get − 10 x , multiply ( x + 5 ) by − 10 x to get − 10 x 2 − 50 x , and subtract from the previous result to get 52 x − 15 .
Divide 52 x by x to get 52 , multiply ( x + 5 ) by 52 to get 52 x + 260 , and subtract from the previous result to get a remainder of − 275 . The final answer is 2 x 2 − 10 x + 52 − x + 5 275 ​ ​ .

Explanation

Understanding the Problem We want to divide the polynomial 2 x 3 + 2 x − 15 by the binomial x + 5 . This is a polynomial division problem, and we can use long division to find the quotient and remainder.

Setting up Long Division First, set up the long division. Write the dividend as 2 x 3 + 0 x 2 + 2 x − 15 to include all powers of x . The divisor is x + 5 .

First Term of Quotient Divide the leading term of the dividend, 2 x 3 , by the leading term of the divisor, x . This gives 2 x 2 . Write 2 x 2 as the first term of the quotient.

Subtracting First Term Multiply the divisor, x + 5 , by 2 x 2 to get 2 x 3 + 10 x 2 . Subtract this from the dividend: ( 2 x 3 + 0 x 2 + 2 x − 15 ) − ( 2 x 3 + 10 x 2 ) = − 10 x 2 + 2 x − 15

Second Term of Quotient Divide the leading term of the new dividend, − 10 x 2 , by the leading term of the divisor, x . This gives − 10 x . Write − 10 x as the next term of the quotient.

Subtracting Second Term Multiply the divisor, x + 5 , by − 10 x to get − 10 x 2 − 50 x . Subtract this from the new dividend: ( − 10 x 2 + 2 x − 15 ) − ( − 10 x 2 − 50 x ) = 52 x − 15

Third Term of Quotient Divide the leading term of the new dividend, 52 x , by the leading term of the divisor, x . This gives 52 . Write 52 as the next term of the quotient.

Subtracting Third Term Multiply the divisor, x + 5 , by 52 to get 52 x + 260 . Subtract this from the new dividend: ( 52 x − 15 ) − ( 52 x + 260 ) = − 275

Final Result The remainder is − 275 . The quotient is 2 x 2 − 10 x + 52 . Therefore, the result of the division is: 2 x 2 − 10 x + 52 − x + 5 275 ​


Examples
Polynomial division is a fundamental concept in algebra and has numerous real-world applications. For instance, engineers use polynomial division to analyze and design control systems. Imagine designing a cruise control system for a car. The system's behavior can be modeled using polynomials, and polynomial division helps engineers determine the system's stability and response to different inputs. By dividing the characteristic polynomial of the system by a factor representing a desired response, they can find the remaining polynomial, which reveals crucial information about the system's performance. This ensures the cruise control smoothly maintains the set speed without oscillations or instability. The result of the division is 2 x 2 − 10 x + 52 − x + 5 275 ​ .

Answered by GinnyAnswer | 2025-07-04