Choose the correct simplification and demonstration of the closure property given: $\left(2 x^3+x^2-4 x\right)-\left(9 x^3-3 x^2\right)$.
A. $-7 x^3+4 x^2-4 x$; may or may not be a polynomial
B. $-7 x^3+4 x^2-4 x$; is a polynomial
C. $11 x^3-2 x^2-4 x$; may or may not be a polynomial
D. $11 x^3-2 x^2-4 x ;$ is a polynomial
Answer (2)
Distribute the negative sign: ( 2 x 3 + x 2 − 4 x ) − ( 9 x 3 − 3 x 2 ) = 2 x 3 + x 2 − 4 x − 9 x 3 + 3 x 2 .
Combine the x 3 terms: 2 x 3 − 9 x 3 = − 7 x 3 .
Combine the x 2 terms: x 2 + 3 x 2 = 4 x 2 .
State the simplified polynomial: − 7 x 3 + 4 x 2 − 4 x ; is a polynomial.
Explanation
Understanding the Problem We are given the expression ( 2 x 3 + x 2 − 4 x ) − ( 9 x 3 − 3 x 2 ) and asked to simplify it and determine if it demonstrates the closure property. The closure property, in this case, means that subtracting two polynomials results in another polynomial.
Expanding the Expression First, we simplify the expression by combining like terms. We have: ( 2 x 3 + x 2 − 4 x ) − ( 9 x 3 − 3 x 2 ) = 2 x 3 + x 2 − 4 x − 9 x 3 + 3 x 2
Combining Like Terms Now, we combine the x 3 terms: 2 x 3 − 9 x 3 = − 7 x 3 .
Next, we combine the x 2 terms: x 2 + 3 x 2 = 4 x 2 .
Finally, the x term is − 4 x .
Simplified Expression So, the simplified expression is − 7 x 3 + 4 x 2 − 4 x .
Identifying the Polynomial Now, we need to determine if the simplified expression is a polynomial. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Since the simplified expression − 7 x 3 + 4 x 2 − 4 x only contains these operations, it is a polynomial.
Conclusion Since the result of subtracting two polynomials is also a polynomial, the closure property is demonstrated. Therefore, the correct answer is − 7 x 3 + 4 x 2 − 4 x ; is a polynomial.
Examples
Polynomials are used to model curves and shapes in engineering and computer graphics. For example, the trajectory of a projectile can be modeled using a quadratic polynomial. Understanding polynomial operations like subtraction helps engineers predict the path of objects or design smooth curves for roads and bridges. Also, in economics, polynomials can represent cost and revenue functions, aiding in business decisions.
The expression ( 2 x 3 + x 2 − 4 x ) − ( 9 x 3 − 3 x 2 ) simplifies to − 7 x 3 + 4 x 2 − 4 x , which is indeed a polynomial. Therefore, the correct choice is − 7 x 3 + 4 x 2 − 4 x ; is a polynomial. This demonstrates the closure property of polynomials.
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