Divide the polynomials by using long division.
1) $(2p^3 + 4p^2 - 24p + 19) \div (p - 2)$
A) $2p^2 + 8p - 5, R -2$
B) $2p^2 + 8p - 5, R 6$
C) $2p^2 + 8p - 10, R 5$
D) $2p^2 + 8p - 8, R 3$
E) $2p^2 + 8p - 10, R 3$
2) $\frac{m^4 - m^3 - 6m^2 - 2m + 2}{m + 1}$
A) $m^3 - 2m^2 - 4m + 2$
B) $m^3 - 2m^2 - 4m + 4$
C) $m^3 - 2m^2 - 2m - 1$
D) $m^3 - 2m^2 - 2m$
E) $m^3 - 2m^2 - 4m + 5$
Answer (2)
Perform polynomial long division for problem 5: $(2p^3 + 4p^2 - 24p + 19)
div (p-2)$. The quotient is 2 p 2 + 8 p − 8 and the remainder is 3 .
The correct answer for problem 5 is D) 2 p 2 + 8 p − 8 , R 3 .
Perform polynomial long division for problem 6: $(m^4 - m^3 - 6m^2 - 2m + 2)
div (m+1) . T h e q u o t i e n t i s m^3 - 2m^2 - 4m + 2$ and the remainder is 0 .
The correct answer for problem 6 is A) m 3 − 2 m 2 − 4 m + 2 .
Explanation
Problem Overview We are asked to divide two polynomials using long division. We have two problems to solve.
Problem 5 Setup For problem 5, we need to divide ( 2 p 3 + 4 p 2 − 24 p + 19 ) by ( p − 2 ) . Let's perform polynomial long division.
First Division Step Dividing 2 p 3 by p gives 2 p 2 . Multiplying ( p − 2 ) by 2 p 2 gives 2 p 3 − 4 p 2 . Subtracting this from 2 p 3 + 4 p 2 gives 8 p 2 . Bringing down the − 24 p gives 8 p 2 − 24 p .
Second Division Step Dividing 8 p 2 by p gives 8 p . Multiplying ( p − 2 ) by 8 p gives 8 p 2 − 16 p . Subtracting this from 8 p 2 − 24 p gives − 8 p . Bringing down the + 19 gives − 8 p + 19 .
Third Division Step Dividing − 8 p by p gives − 8 . Multiplying ( p − 2 ) by − 8 gives − 8 p + 16 . Subtracting this from − 8 p + 19 gives 3 . Therefore, the quotient is 2 p 2 + 8 p − 8 and the remainder is 3 .
Problem 5 Answer Comparing our result with the multiple choice options, we see that the correct answer for problem 5 is D) 2 p 2 + 8 p − 8 , R 3 .
Problem 6 Setup For problem 6, we need to divide ( m 4 − m 3 − 6 m 2 − 2 m + 2 ) by ( m + 1 ) . Let's perform polynomial long division.
First Division Step Dividing m 4 by m gives m 3 . Multiplying ( m + 1 ) by m 3 gives m 4 + m 3 . Subtracting this from m 4 − m 3 gives − 2 m 3 . Bringing down the − 6 m 2 gives − 2 m 3 − 6 m 2 .
Second Division Step Dividing − 2 m 3 by m gives − 2 m 2 . Multiplying ( m + 1 ) by − 2 m 2 gives − 2 m 3 − 2 m 2 . Subtracting this from − 2 m 3 − 6 m 2 gives − 4 m 2 . Bringing down the − 2 m gives − 4 m 2 − 2 m .
Third Division Step Dividing − 4 m 2 by m gives − 4 m . Multiplying ( m + 1 ) by − 4 m gives − 4 m 2 − 4 m . Subtracting this from − 4 m 2 − 2 m gives 2 m . Bringing down the + 2 gives 2 m + 2 .
Fourth Division Step Dividing 2 m by m gives 2 . Multiplying ( m + 1 ) by 2 gives 2 m + 2 . Subtracting this from 2 m + 2 gives 0 . Therefore, the quotient is m 3 − 2 m 2 − 4 m + 2 and the remainder is 0 .
Problem 6 Answer Comparing our result with the multiple choice options, we see that the correct answer for problem 6 is A) m 3 − 2 m 2 − 4 m + 2 .
Examples
Polynomial long division is a fundamental technique in algebra, useful in various real-world applications. For instance, engineers use polynomial division to analyze and design control systems. Imagine designing a cruise control system for a car; the system's behavior can be modeled using polynomials, and division helps determine stability and response characteristics. Similarly, in signal processing, polynomial division is used to decode and filter signals, ensuring clear communication. These applications highlight the practical importance of mastering polynomial division.
The quotient for the first problem is 2 p 2 + 8 p − 8 with a remainder of 3 (Answer D), and for the second problem, the quotient is m 3 − 2 m 2 − 4 m + 2 with no remainder (Answer A).
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