Divide $x^2+5x+6$ by $x+3$.
Solve: $(6x^2+5x+1) \div (x+2)$
Answer (2)
Divide x 2 + 5 x + 6 by x + 3 by factoring x 2 + 5 x + 6 into ( x + 2 ) ( x + 3 ) , then cancel out ( x + 3 ) to get x + 2 .
Divide 6 x 2 + 5 x + 1 by x + 2 using polynomial long division to get 6 x − 7 + x + 2 15 .
The result of dividing x 2 + 5 x + 6 by x + 3 is x + 2 .
The result of dividing 6 x 2 + 5 x + 1 by x + 2 is 6 x − 7 + x + 2 15 .
Explanation
Understanding the Problem We are asked to divide two polynomials. First, we need to divide x 2 + 5 x + 6 by x + 3 , and then we need to divide 6 x 2 + 5 x + 1 by x + 2 .
Dividing the First Polynomial Let's start by dividing x 2 + 5 x + 6 by x + 3 . We can use polynomial long division or factoring. Let's try factoring the quadratic 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, we can factor the quadratic as ( x + 2 ) ( x + 3 ) . Therefore, x + 3 x 2 + 5 x + 6 = x + 3 ( x + 2 ) ( x + 3 ) = x + 2 .
Dividing the Second Polynomial Now, let's divide 6 x 2 + 5 x + 1 by x + 2 . We can use polynomial long division.
We set up the long division as follows:
6x - 7
x + 2 | 6x^2 + 5x + 1 -(6x^2 + 12x) ---------------- -7x + 1 -(-7x - 14) ---------------- 15
So, 6 x 2 + 5 x + 1 = ( x + 2 ) ( 6 x − 7 ) + 15 . Therefore, x + 2 6 x 2 + 5 x + 1 = 6 x − 7 + x + 2 15 .
Final Answer Therefore, the result of dividing x 2 + 5 x + 6 by x + 3 is x + 2 , and the result of dividing 6 x 2 + 5 x + 1 by x + 2 is 6 x − 7 + x + 2 15 .
Examples
Polynomial division is used in many engineering and scientific applications. For example, when designing a bridge, engineers use polynomial functions to model the load distribution. Dividing these polynomials helps them determine the stress and strain at different points on the bridge, ensuring its stability and safety. Similarly, in signal processing, polynomial division is used to analyze and filter signals, removing unwanted noise and improving the clarity of the desired information. This technique is also fundamental in control systems, where it helps in designing controllers that stabilize and optimize the performance of dynamic systems.
To divide x 2 + 5 x + 6 by x + 3 , we factor it and find the result is x + 2 . For 6 x 2 + 5 x + 1 divided by x + 2 , using polynomial long division gives us 6 x − 7 + x + 2 15 .
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