Use long division to find the quotient below.
$\left(16 x^3+4 x^2-144\right)+(4 x-8)$
A. $4 x^2-15 x+18$
B. $4 x^2+15 x+18$
C. $4 x^2-9 x+18$
D. $4 x^2+9 x+18$
Answer (1)
Divide 16 x 3 by 4 x to get the first term of the quotient: 4 x 2 .
Multiply ( 4 x − 8 ) by 4 x 2 and subtract from the dividend to get 36 x 2 + 0 x .
Divide 36 x 2 by 4 x to get the second term of the quotient: 9 x .
Multiply ( 4 x − 8 ) by 9 x and subtract from the dividend to get 72 x − 144 .
Divide 72 x by 4 x to get the third term of the quotient: 18 .
Multiply ( 4 x − 8 ) by 18 and subtract from the dividend to get a remainder of 0. The quotient is 4 x 2 + 9 x + 18 .
Explanation
Understanding the Problem We are asked to perform polynomial long division to find the quotient of ( 16 x 3 + 4 x 2 − 144 ) divided by ( 4 x − 8 ) . This means we want to find a polynomial Q ( x ) such that ( 16 x 3 + 4 x 2 − 144 ) = ( 4 x − 8 ) Q ( x ) + R ( x ) , where R ( x ) is the remainder. In this case, we expect the remainder to be 0.
First Term of Quotient First, we set up the long division. We write the dividend as 16 x 3 + 4 x 2 + 0 x − 144 to include all powers of x . Then we divide the first term of the dividend, 16 x 3 , by the first term of the divisor, 4 x . This gives us 4 x 16 x 3 = 4 x 2 . This is the first term of the quotient.
Multiply and Subtract Next, we multiply the divisor ( 4 x − 8 ) by 4 x 2 to get 4 x 2 ( 4 x − 8 ) = 16 x 3 − 32 x 2 . We subtract this from the dividend: ( 16 x 3 + 4 x 2 ) − ( 16 x 3 − 32 x 2 ) = 36 x 2 . Then we bring down the next term, 0 x , to get 36 x 2 + 0 x .
Second Term of Quotient Now, we divide the first term of the new dividend, 36 x 2 , by the first term of the divisor, 4 x . This gives us 4 x 36 x 2 = 9 x . This is the second term of the quotient.
Multiply and Subtract We multiply the divisor ( 4 x − 8 ) by 9 x to get 9 x ( 4 x − 8 ) = 36 x 2 − 72 x . We subtract this from the new dividend: ( 36 x 2 + 0 x ) − ( 36 x 2 − 72 x ) = 72 x . Then we bring down the next term, − 144 , to get 72 x − 144 .
Third Term of Quotient Next, we divide the first term of the new dividend, 72 x , by the first term of the divisor, 4 x . This gives us 4 x 72 x = 18 . This is the third term of the quotient.
Multiply and Subtract We multiply the divisor ( 4 x − 8 ) by 18 to get 18 ( 4 x − 8 ) = 72 x − 144 . We subtract this from the new dividend: ( 72 x − 144 ) − ( 72 x − 144 ) = 0 . The remainder is 0.
Final Answer Therefore, the quotient is 4 x 2 + 9 x + 18 .
Examples
Polynomial long division is a fundamental technique in algebra, useful in various applications such as simplifying complex rational expressions, solving polynomial equations, and analyzing the behavior of polynomial functions. For instance, engineers might use polynomial division to analyze the stability of control systems, where the characteristic equation of the system is a polynomial. By dividing one polynomial by another, they can simplify the equation and determine the system's stability criteria. This method is also crucial in cryptography for tasks like error detection and correction in data transmission, ensuring data integrity.
