What is the quotient of $(x^3+6 x^2+11 x+6) \div (x^2+4 x+3)$?
A. $x+2$
B. $x-2$
C. $x+10$
D. $x+6$
Answer (1)
Factor the numerator: x 3 + 6 x 2 + 11 x + 6 = ( x + 1 ) ( x + 2 ) ( x + 3 ) .
Factor the denominator: x 2 + 4 x + 3 = ( x + 1 ) ( x + 3 ) .
Divide the numerator by the denominator: ( x + 1 ) ( x + 3 ) ( x + 1 ) ( x + 2 ) ( x + 3 ) .
Cancel common factors to find the quotient: x + 2 .
Explanation
Problem Analysis We are asked to find the quotient of the polynomial division x 2 + 4 x + 3 x 3 + 6 x 2 + 11 x + 6 . To do this, we will first factor both the numerator and the denominator, and then simplify the expression.
Factoring the Numerator Let's factor the numerator, x 3 + 6 x 2 + 11 x + 6 . We look for integer roots by checking factors of the constant term, 6. We find that x = − 1 is a root, since ( − 1 ) 3 + 6 ( − 1 ) 2 + 11 ( − 1 ) + 6 = − 1 + 6 − 11 + 6 = 0 . Thus, ( x + 1 ) is a factor. We can perform polynomial long division or synthetic division to find the other factor. Alternatively, we can look for two numbers that multiply to 6 and add up to 6-1 = 5. These numbers are 2 and 3. Thus, we can write the numerator as ( x + 1 ) ( x + 2 ) ( x + 3 ) .
Factoring the Denominator Now, let's factor the denominator, x 2 + 4 x + 3 . We look for two numbers that multiply to 3 and add up to 4. These numbers are 1 and 3. Thus, we can write the denominator as ( x + 1 ) ( x + 3 ) .
Simplifying the Expression Now we can write the division as: x 2 + 4 x + 3 x 3 + 6 x 2 + 11 x + 6 = ( x + 1 ) ( x + 3 ) ( x + 1 ) ( x + 2 ) ( x + 3 ) We can cancel the common factors ( x + 1 ) and ( x + 3 ) from the numerator and the denominator, which gives us x + 2 .
Final Answer Therefore, the quotient of the given expression is x + 2 .
Examples
Polynomial division is used in various engineering and scientific applications. For example, when designing a bridge, engineers use polynomial functions to model the load distribution. Simplifying these functions through division helps in determining the structural integrity and safety margins. Similarly, in signal processing, polynomial division is used to analyze and filter signals, ensuring clear communication and accurate data transmission. Understanding polynomial division enables professionals to optimize designs and improve the performance of systems in real-world scenarios.
