What is the quotient of $(x^3+3 x^2-4 x-12) \div (x^2+5 x+6)$?

Factor the numerator x 3 + 3 x 2 − 4 x − 12 by grouping: ( x + 3 ) ( x − 2 ) ( x + 2 ) .
Factor the denominator x 2 + 5 x + 6 : ( x + 2 ) ( x + 3 ) .
Simplify the rational expression ( x + 2 ) ( x + 3 ) ( x + 3 ) ( x − 2 ) ( x + 2 ) ​ by canceling common factors.
The quotient is x − 2 ​ .

Explanation

Understanding the Problem We are asked to find the quotient of two polynomials: x 2 + 5 x + 6 x 3 + 3 x 2 − 4 x − 12 ​ . This means we need to divide the first polynomial by the second polynomial. To do this, we will first factor both polynomials and then simplify the resulting rational expression.

Factoring the Numerator Let's factor the numerator, x 3 + 3 x 2 − 4 x − 12 . We can use factoring by grouping. We group the first two terms and the last two terms: ( x 3 + 3 x 2 ) + ( − 4 x − 12 ) . From the first group, we can factor out x 2 , and from the second group, we can factor out − 4 . This gives us x 2 ( x + 3 ) − 4 ( x + 3 ) . Now we can factor out the common factor of ( x + 3 ) to get ( x + 3 ) ( x 2 − 4 ) . Finally, we can factor the difference of squares x 2 − 4 as ( x − 2 ) ( x + 2 ) . So the numerator factors as ( x + 3 ) ( x − 2 ) ( x + 2 ) .

Factoring the Denominator Now let's factor the denominator, x 2 + 5 x + 6 . We are looking for two numbers that multiply to 6 and add to 5. These numbers are 2 and 3. So the denominator factors as ( x + 2 ) ( x + 3 ) .

Simplifying the Rational Expression Now we can write the rational expression as ( x + 2 ) ( x + 3 ) ( x + 3 ) ( x − 2 ) ( x + 2 ) ​ . We can cancel the common factors of ( x + 3 ) and ( x + 2 ) from the numerator and the denominator, which gives us x − 2 .

Finding the Quotient Therefore, the quotient of x 2 + 5 x + 6 x 3 + 3 x 2 − 4 x − 12 ​ is x − 2 .


Examples
Understanding polynomial division is crucial in various fields, such as engineering and computer science. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. Dividing one polynomial by another helps them determine the stress at different points. Similarly, in computer graphics, polynomial division is used to manipulate curves and surfaces, allowing for realistic rendering of 3D objects. This algebraic skill enables precise control and optimization in complex systems.

Answered by GinnyAnswer | 2025-07-05

The quotient of ( x 3 + 3 x 2 − 4 x − 12 ) ÷ ( x 2 + 5 x + 6 ) is found by factoring both the numerator and denominator, resulting in a simplified form of x − 2 . Thus, the final answer is x − 2 ​ .
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Answered by Anonymous | 2025-07-07