Use long division to find the quotient below.
$\left(15 x^3+2 x^2-75\right)-(3 x-5)$
A. $5 x^2+9 x+15$
B. $5 x^2+12 x+15$
C. $5 x^2-12 x+15$
D. $5 x^2-9 x+15$
Answer (1)
Set up the polynomial long division with dividend 15 x 3 + 2 x 2 − 75 and divisor 3 x − 5 .
Divide 15 x 3 by 3 x to get the first term of the quotient: 5 x 2 .
Multiply ( 3 x − 5 ) by 5 x 2 and subtract from the dividend, resulting in 27 x 2 − 75 .
Divide 27 x 2 by 3 x to get the second term of the quotient: 9 x .
Multiply ( 3 x − 5 ) by 9 x and subtract, resulting in 45 x − 75 .
Divide 45 x by 3 x to get the third term of the quotient: 15 .
Multiply ( 3 x − 5 ) by 15 and subtract, resulting in a remainder of 0 .
The quotient is 5 x 2 + 9 x + 15 .
Explanation
Understanding the Problem We are asked to perform polynomial long division to find the quotient of ( 15 x 3 + 2 x 2 − 75 ) divided by ( 3 x − 5 ) . This means we want to find a polynomial Q ( x ) such that ( 15 x 3 + 2 x 2 − 75 ) = ( 3 x − 5 ) Q ( x ) + R ( x ) , where R ( x ) is the remainder. In this case, we expect the remainder to be 0.
Setting up Long Division First, set up the long division problem. Write the dividend as 15 x 3 + 2 x 2 + 0 x − 75 to include all powers of x .
Finding the First Term of the Quotient Divide the first term of the dividend, 15 x 3 , by the first term of the divisor, 3 x . This gives us 3 x 15 x 3 = 5 x 2 . This is the first term of the quotient.
Multiplying the Divisor Multiply the divisor, 3 x − 5 , by 5 x 2 to get 5 x 2 ( 3 x − 5 ) = 15 x 3 − 25 x 2 .
Subtracting and Bringing Down Subtract this result from the dividend: ( 15 x 3 + 2 x 2 ) − ( 15 x 3 − 25 x 2 ) = 27 x 2 . Bring down the next term, 0 x , to get 27 x 2 + 0 x .
Finding the Second Term of the Quotient Divide the first term of the new dividend, 27 x 2 , by the first term of the divisor, 3 x . This gives us 3 x 27 x 2 = 9 x . This is the second term of the quotient.
Multiplying the Divisor Multiply the divisor, 3 x − 5 , by 9 x to get 9 x ( 3 x − 5 ) = 27 x 2 − 45 x .
Subtracting and Bringing Down Subtract this result from the new dividend: ( 27 x 2 + 0 x ) − ( 27 x 2 − 45 x ) = 45 x . Bring down the next term, − 75 , to get 45 x − 75 .
Finding the Third Term of the Quotient Divide the first term of the new dividend, 45 x , by the first term of the divisor, 3 x . This gives us 3 x 45 x = 15 . This is the third term of the quotient.
Multiplying the Divisor Multiply the divisor, 3 x − 5 , by 15 to get 15 ( 3 x − 5 ) = 45 x − 75 .
Final Subtraction Subtract this result from the new dividend: ( 45 x − 75 ) − ( 45 x − 75 ) = 0 . The remainder is 0.
The Quotient The quotient is 5 x 2 + 9 x + 15 .
Examples
Polynomial long division is a method used to divide a polynomial by another polynomial of lower or equal degree. For example, if you are designing a rectangular garden where the area is represented by the polynomial 15 x 3 + 2 x 2 − 75 and one side is represented by 3 x − 5 , polynomial long division helps you find the expression for the other side, which in this case is 5 x 2 + 9 x + 15 . This technique is also fundamental in simplifying complex algebraic expressions and solving higher-degree polynomial equations, which are essential in fields like engineering, physics, and computer science for modeling and solving real-world problems.
