Divide the following: $\frac{4 x^3+3 x^2+2 x-3}{x^2-2}$
Answer (1)
Perform polynomial long division to divide 4 x 3 + 3 x 2 + 2 x − 3 by x 2 − 2 .
Divide 4 x 3 by x 2 to get the first term of the quotient, 4 x .
Multiply 4 x by x 2 − 2 and subtract the result from the dividend.
Divide 3 x 2 by x 2 to get the next term of the quotient, 3 .
Multiply 3 by x 2 − 2 and subtract the result from the previous remainder.
The quotient is 4 x + 3 and the remainder is 10 x + 3 , so the final answer is 4 x + 3 + x 2 − 2 10 x + 3 .
Explanation
Understanding the Problem We are asked to divide the polynomial 4 x 3 + 3 x 2 + 2 x − 3 by x 2 − 2 . Since the denominator is a polynomial of degree 2, we will use long division to solve this problem.
Setting up Long Division We set up the long division as follows:
\t\t\t\t\t x 2 − 2 ) 4 x 3 + 3 x 2 + 2 x − 3
First Term of Quotient First, we divide the leading term of the dividend, 4 x 3 , by the leading term of the divisor, x 2 . This gives us 4 x . We write this as the first term of the quotient and multiply the divisor by 4 x :
4 x ( x 2 − 2 ) = 4 x 3 − 8 x .
Subtracting We subtract this from the dividend:
( 4 x 3 + 3 x 2 + 2 x − 3 ) − ( 4 x 3 − 8 x ) = 3 x 2 + 10 x − 3 .
Second Term of Quotient Now, we divide the leading term of the new dividend, 3 x 2 , by the leading term of the divisor, x 2 . This gives us 3 . We write this as the next term of the quotient and multiply the divisor by 3 :
3 ( x 2 − 2 ) = 3 x 2 − 6 .
Subtracting Again We subtract this from the new dividend:
( 3 x 2 + 10 x − 3 ) − ( 3 x 2 − 6 ) = 10 x + 3 .
Final Result Since the degree of the remainder, 10 x + 3 , is less than the degree of the divisor, x 2 − 2 , we stop the long division. The quotient is 4 x + 3 and the remainder is 10 x + 3 .
Therefore, x 2 − 2 4 x 3 + 3 x 2 + 2 x − 3 = 4 x + 3 + x 2 − 2 10 x + 3 .
Conclusion Thus, the result of the division is 4 x + 3 with a remainder of 10 x + 3 .
Examples
Polynomial long division is a fundamental technique in algebra, useful in various applications. For instance, engineers use polynomial division to analyze control systems, where the transfer function of a system is often expressed as a ratio of polynomials. By performing polynomial division, they can simplify the transfer function and understand the system's behavior more easily. Similarly, in computer graphics, polynomial division can be used to manipulate curves and surfaces, allowing for efficient rendering and modeling.
