Determine if the expression $-\frac{7}{m^3}$ is a polynomial or not. If it is a polynomial, state the type and degree of polynomial.
Answer (2)
Rewrite the expression as − 7 m − 3 .
Recall that polynomials only have non-negative integer exponents.
Since the exponent of m is -3, the expression is not a polynomial.
Therefore, the expression is not a polynomial.
Explanation
Understanding the Problem We are given the expression − m 3 7 and we need to determine if it is a polynomial. If it is, we need to state its type and degree.
Rewriting the Expression First, let's rewrite the expression using a negative exponent: − m 3 7 = − 7 m − 3
Recalling the Definition of a Polynomial Recall the definition of 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.
Checking the Exponent In our expression, the exponent of the variable 'm' is -3, which is a negative integer. According to the definition of a polynomial, the exponents of the variables must be non-negative integers.
Conclusion Since the exponent of 'm' is negative, the given expression is not 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. Understanding polynomials helps in making predictions and solving problems in physics, engineering, and economics. For example, if you throw a ball, the height of the ball over time can be modeled by a quadratic polynomial. Similarly, the cost of producing a certain number of items can be modeled by a polynomial function.
The expression − m 3 7 is not a polynomial because it can be rewritten as − 7 m − 3 , where the exponent -3 is negative. Polynomials can only have non-negative integer exponents. Thus, this expression does not meet the definition of a polynomial.
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