Choose the correct simplification of $\left(7 x^3-8 x-5\right)+\left(3 x^3+7 x+1\right)$.
A. $10 x^3-x-4$
B. $10 x^3+x+4$
C. $4 x^3-15 x-6$
D. $4 x^3+15 x+6$
Answer (2)
Combine the x 3 terms: 7 x 3 + 3 x 3 = 10 x 3 .
Combine the x terms: − 8 x + 7 x = − x .
Combine the constant terms: − 5 + 1 = − 4 .
Write the simplified expression: 10 x 3 − x − 4 .
Explanation
Understanding the Problem We are given the expression ( 7 x 3 − 8 x − 5 ) + ( 3 x 3 + 7 x + 1 ) and asked to simplify it. This involves combining like terms, which are terms with the same variable raised to the same power. We will add the coefficients of the x 3 terms, the x terms, and the constant terms separately.
Combining x 3 Terms First, let's combine the x 3 terms. We have 7 x 3 and 3 x 3 . Adding their coefficients, we get 7 + 3 = 10 . So, the combined term is 10 x 3 .
Combining x Terms Next, let's combine the x terms. We have − 8 x and 7 x . Adding their coefficients, we get − 8 + 7 = − 1 . So, the combined term is − 1 x , which we can write as − x .
Combining Constant Terms Finally, let's combine the constant terms. We have − 5 and 1 . Adding them, we get − 5 + 1 = − 4 .
Writing the Simplified Expression Now, we write the simplified expression by combining the results from the previous steps: 10 x 3 − x − 4 . Therefore, the correct simplification is 10 x 3 − x − 4 .
Examples
Simplifying polynomial expressions is a fundamental skill in algebra and is used in various real-world applications. For example, when designing structures like bridges or buildings, engineers use polynomial expressions to model the forces and stresses acting on different parts of the structure. By simplifying these expressions, they can more easily analyze the structural integrity and ensure the safety of the design. Similarly, in computer graphics, polynomial expressions are used to represent curves and surfaces, and simplifying these expressions can improve the efficiency of rendering complex scenes.
The simplification of the expression ( 7 x 3 − 8 x − 5 ) + ( 3 x 3 + 7 x + 1 ) is 10 x 3 − x − 4 . Thus, the correct answer is A. 10 x 3 − x − 4 .
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