Choose the correct simplification and demonstration of the closure property given: (4x³ + 3x²-6x) - (10x³ + 3x²).
A. -6x³-6x; is a polynomial
B. -6x³-6x; may or may not be a polynomial
C. -6x³-6x²-6x; is a polynomial
D. -6x³-6x²-6x; may or may not be a polynomial
Answer (2)
Distribute the negative sign: ( 4 x 3 + 3 x 2 − 6 x ) − ( 10 x 3 + 3 x 2 ) = 4 x 3 + 3 x 2 − 6 x − 10 x 3 − 3 x 2 .
Combine the x 3 terms: 4 x 3 − 10 x 3 = − 6 x 3 .
Combine the x 2 terms: 3 x 2 − 3 x 2 = 0 .
The simplified expression is − 6 x 3 − 6 x , which is a polynomial. Therefore, the answer is − 6 x 3 − 6 x ; is a polynomial .
Explanation
Understanding the Problem We are given the expression ( 4 x 3 + 3 x 2 − 6 x ) − ( 10 x 3 + 3 x 2 ) and asked to simplify it and determine if the result is a polynomial.
Simplifying the Expression To simplify the expression, we combine like terms. This means we group together terms with the same power of x .
Distributing the Negative Sign First, let's rewrite the expression by distributing the negative sign: ( 4 x 3 + 3 x 2 − 6 x ) − ( 10 x 3 + 3 x 2 ) = 4 x 3 + 3 x 2 − 6 x − 10 x 3 − 3 x 2
Combining x 3 Terms Now, we combine the x 3 terms: 4 x 3 − 10 x 3 = − 6 x 3
Combining x 2 Terms Next, we combine the x 2 terms: 3 x 2 − 3 x 2 = 0 x 2 = 0
Identifying the x Term The x term is − 6 x , which remains unchanged.
Simplified Expression So, the simplified expression is: − 6 x 3 + 0 x 2 − 6 x = − 6 x 3 − 6 x
Checking if it 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 of variables. Since the simplified expression − 6 x 3 − 6 x only involves integer exponents and the operations of addition, subtraction, and multiplication, it is a polynomial.
Final Answer Therefore, the correct simplification is − 6 x 3 − 6 x , and it is a polynomial.
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a ball, the shape of a bridge, or the growth of a population. Simplifying polynomial expressions allows us to analyze these models more easily and make predictions about the phenomena they represent. For example, if we have a polynomial that represents the profit of a company, simplifying it can help us determine the factors that affect the profit and make decisions to improve it.
The expression ( 4 x 3 + 3 x 2 − 6 x ) − ( 10 x 3 + 3 x 2 ) simplifies to − 6 x 3 − 6 x , which is a polynomial. Therefore, the correct answer is A: − 6 x 3 − 6 x ; is a polynomial .
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