Perform the indicated operations on the following polynomials. Arrange answer in descending powers.
$\left(6 x^3+27 x-19 x^2-15\right) \div(3 x-5)=$
A. $4 x^2-3 x+2+\frac{3}{5 x-5}$
B. $3 x^2-2 x+5+\frac{4}{3 x-5}$
C. $2 x^3-4 x+3+\frac{4}{3 x-5}$
D. $2 x^2-3 x+4+\frac{5}{3 x-5}$
Answer (2)
Rewrite the dividend in descending powers of x: 6 x 3 − 19 x 2 + 27 x − 15 .
Perform polynomial long division.
Identify the quotient and the remainder.
Write the result as 2 x 2 − 3 x + 4 + 3 x − 5 5 .
Explanation
Understanding the Problem We are given the polynomial division problem: ( 6 x 3 + 27 x − 19 x 2 − 15 ) ÷ ( 3 x − 5 ) . Our goal is to perform the division and express the result in descending powers of x .
Rewriting the Dividend First, rewrite the dividend in descending powers of x : 6 x 3 − 19 x 2 + 27 x − 15 . Now, we will perform polynomial long division with 6 x 3 − 19 x 2 + 27 x − 15 as the dividend and 3 x − 5 as the divisor.
First Term of the Quotient The first term of the quotient will be 3 x 6 x 3 = 2 x 2 . Multiply the divisor by 2 x 2 to get 2 x 2 ( 3 x − 5 ) = 6 x 3 − 10 x 2 . Subtract this from the dividend: ( 6 x 3 − 19 x 2 + 27 x − 15 ) − ( 6 x 3 − 10 x 2 ) = − 9 x 2 + 27 x − 15 .
Second Term of the Quotient The next term of the quotient will be 3 x − 9 x 2 = − 3 x . Multiply the divisor by − 3 x to get − 3 x ( 3 x − 5 ) = − 9 x 2 + 15 x . Subtract this from the remaining dividend: ( − 9 x 2 + 27 x − 15 ) − ( − 9 x 2 + 15 x ) = 12 x − 15 .
Third Term of the Quotient The next term of the quotient will be 3 x 12 x = 4 . Multiply the divisor by 4 to get 4 ( 3 x − 5 ) = 12 x − 20 . Subtract this from the remaining dividend: ( 12 x − 15 ) − ( 12 x − 20 ) = 5 . This is the remainder.
Final Result Therefore, the result of the division is 2 x 2 − 3 x + 4 + 3 x − 5 5 .
Examples
Polynomial division is a fundamental concept in algebra with numerous real-world applications. For instance, engineers use polynomial division to model and analyze systems, such as electrical circuits or control systems. By representing system behavior with polynomials, engineers can simplify complex equations and predict system responses. This allows them to design more efficient and reliable systems, ensuring optimal performance and stability.
The division of the polynomial ( 6 x 3 + 27 x − 19 x 2 − 15 ) ÷ ( 3 x − 5 ) yields 2 x 2 − 3 x + 4 + 3 x − 5 5 . The correct answer matches option D. This result is achieved through polynomial long division.
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