$\frac{4 x^3+6 x^2-2 x+3}{x+2}$
Answer (1)
Divide 4 x 3 by x to get 4 x 2 , then multiply ( x + 2 ) by 4 x 2 and subtract from the original polynomial.
Divide − 2 x 2 by x to get − 2 x , then multiply ( x + 2 ) by − 2 x and subtract from the remaining polynomial.
Divide 2 x by x to get 2 , then multiply ( x + 2 ) by 2 and subtract from the remaining polynomial.
The quotient is 4 x 2 − 2 x + 2 and the remainder is − 1 , so the final answer is 4 x 2 − 2 x + 2 − x + 2 1 .
Explanation
Understanding the Problem We are asked to divide the polynomial 4 x 3 + 6 x 2 − 2 x + 3 by x + 2 . We will use polynomial long division to find the quotient and remainder.
First Step of Long Division First, divide the leading term of the dividend ( 4 x 3 ) by the leading term of the divisor ( x ). This gives us 4 x 2 . Multiply the divisor ( x + 2 ) by 4 x 2 to get 4 x 3 + 8 x 2 . Subtract this from the dividend to get a new dividend.
Subtracting the Terms The calculation is as follows:
4x^3 + 6x^2 - 2x + 3 - (4x^3 + 8x^2) = -2x^2 - 2x + 3
Second Step of Long Division Now, divide the leading term of the new dividend ( − 2 x 2 ) by the leading term of the divisor ( x ). This gives us − 2 x . Multiply the divisor ( x + 2 ) by − 2 x to get − 2 x 2 − 4 x . Subtract this from the new dividend to get another new dividend.
Subtracting the Terms The calculation is as follows:
-2x^2 - 2x + 3 - (-2x^2 - 4x) = 2x + 3
Third Step of Long Division Next, divide the leading term of the new dividend ( 2 x ) by the leading term of the divisor ( x ). This gives us 2 . Multiply the divisor ( x + 2 ) by 2 to get 2 x + 4 . Subtract this from the new dividend to get the remainder.
Subtracting the Terms The calculation is as follows:
2x + 3 - (2x + 4) = -1
Final Result The quotient is 4 x 2 − 2 x + 2 and the remainder is − 1 . Therefore, we can write the result as:
x + 2 4 x 3 + 6 x 2 − 2 x + 3 = 4 x 2 − 2 x + 2 − x + 2 1
Conclusion Thus, the result of the division is 4 x 2 − 2 x + 2 with a remainder of − 1 .
Examples
Polynomial long division is used in various applications, such as simplifying complex rational expressions in calculus, designing control systems in engineering, and optimizing resource allocation in economics. For example, when designing a chemical reactor, engineers use polynomial division to analyze the concentration of reactants and products over time. By simplifying the rate equations, they can predict the reactor's performance and optimize its efficiency. This ensures that the chemical process operates safely and economically.
