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 ( x 2 − 2 ) by 4 x and subtract from the dividend to get 3 x 2 + 10 x − 3 .
Divide 3 x 2 by x 2 to get the second term of the quotient: 3 .
Multiply ( x 2 − 2 ) by 3 and subtract from the new dividend to get the remainder 10 x + 3 .
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 polynomial long division to solve this problem.
First Step of Long Division We set up the long division as follows:
x^2 - 2 | 4x^3 + 3x^2 + 2x - 3
We divide the leading term of the dividend ( 4 x 3 ) by the leading term of the divisor ( x 2 ) to get the first term of the quotient, which is 4 x .
Subtracting the First Term We multiply the divisor ( x 2 − 2 ) by 4 x to get 4 x 3 − 8 x . Then we subtract this from the dividend:
4x
x^2 - 2 | 4x^3 + 3x^2 + 2x - 3
-(4x^3 - 8x)
------------------
3x^2 + 10x - 3
Finding the Second Term Now we divide the leading term of the new dividend ( 3 x 2 ) by the leading term of the divisor ( x 2 ) to get the next term of the quotient, which is 3 .
Subtracting the Second Term We multiply the divisor ( x 2 − 2 ) by 3 to get 3 x 2 − 6 . Then we subtract this from the new dividend:
4x + 3
x^2 - 2 | 4x^3 + 3x^2 + 2x - 3
-(4x^3 - 8x)
------------------
3x^2 + 10x - 3
-(3x^2 - 6)
------------------
10x + 3
Final Result The remainder is 10 x + 3 , and the quotient is 4 x + 3 . Therefore, we can write the result as:
x 2 − 2 4 x 3 + 3 x 2 + 2 x − 3 = 4 x + 3 + x 2 − 2 10 x + 3
So the quotient is 4 x + 3 and the remainder is 10 x + 3 .
Examples
Polynomial long division is a fundamental technique in algebra, useful in various applications such as simplifying complex rational expressions and solving polynomial equations. For instance, consider designing a rectangular garden where the area is represented by the polynomial 4 x 3 + 3 x 2 + 2 x − 3 square feet. If one side of the garden is represented by x 2 − 2 feet, polynomial long division helps determine the expression for the other side, ensuring efficient space utilization and garden layout.
