What is the square root of $(x+1)(x+2)(x+3)(x+4)+1$?
A. $x^2+4 x+4$
B. $x^2+5 x+5$
C. $x^2+5$
D. $x^2+6 x+6$
Answer (2)
Rearrange the terms: ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + 1 = ( x 2 + 5 x + 4 ) ( x 2 + 5 x + 6 ) + 1 .
Substitute y = x 2 + 5 x : The expression becomes ( y + 4 ) ( y + 6 ) + 1 = ( y + 5 ) 2 .
Take the square root: ( y + 5 ) 2 = ∣ y + 5∣ = ∣ x 2 + 5 x + 5∣ .
The square root is x 2 + 5 x + 5 .
Explanation
Understanding the Problem We are asked to find the square root of the expression ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + 1 . The answer should be one of the given options.
Rearranging the terms To simplify the expression, we rearrange the terms in the product as follows: ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + 1 = ( x + 1 ) ( x + 4 ) ( x + 2 ) ( x + 3 ) + 1 = ( x 2 + 5 x + 4 ) ( x 2 + 5 x + 6 ) + 1.
Substitution Let y = x 2 + 5 x . Then the expression becomes ( y + 4 ) ( y + 6 ) + 1 = y 2 + 10 y + 24 + 1 = y 2 + 10 y + 25 = ( y + 5 ) 2 .
Finding the Square Root Therefore, ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + 1 = ( y + 5 ) 2 = ∣ y + 5∣ = ∣ x 2 + 5 x + 5∣.
Analyzing the Expression Since we are looking for a polynomial expression, we consider the expression inside the absolute value. We have x 2 + 5 x + 5 . To determine if this expression is always positive, we can find its roots. Using the quadratic formula, the roots are x = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( 5 ) = 2 − 5 ± 5 . These roots are approximately -1.38 and -3.62. Since the parabola opens upwards (the coefficient of x 2 is positive), the expression is negative between the roots and positive outside the roots. However, the options provided do not include absolute value. So we assume the expression inside the absolute value is positive for the domain we are considering. Therefore, the square root is x 2 + 5 x + 5 .
Final Answer Comparing the result with the given options, the correct option is b. x 2 + 5 x + 5 .
Examples
This type of problem, involving polynomial manipulation and simplification, is useful in various fields such as engineering and computer science. For example, when designing algorithms or control systems, engineers often need to simplify complex polynomial expressions to make them easier to analyze and implement. By recognizing patterns and using algebraic techniques, they can optimize their designs and improve performance. Similarly, in computer graphics, simplifying polynomial expressions can help to speed up rendering and improve the visual quality of images.
The square root of the expression ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + 1 simplifies to x 2 + 5 x + 5 . Therefore, the correct answer is option B. x 2 + 5 x + 5 .
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