Determine if the expression $-a-10 a^2 \sqrt{5}+8 \sqrt{a}$ is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial.
Answer (2)
A polynomial consists of variables and coefficients with non-negative integer exponents.
Examine the expression − a − 10 a 2 5 + 8 a .
The term 8 a = 8 a 2 1 has a non-integer exponent.
Therefore, the expression is not a polynomial: Not a polynomial .
Explanation
Understanding the Problem We are given the expression − a − 10 a 2 5 + 8 a and asked to determine if it is a polynomial. If it is, we need to state its type and degree.
Definition of a Polynomial Recall that a polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of the variables.
Analyzing Each Term Let's examine each term in the given expression:
The first term is − a , which can be written as − 1 ⋅ a 1 . The exponent of a is 1, which is a non-negative integer. So, this term is a polynomial term.
The second term is − 10 a 2 5 , which can be written as − 10 5 ⋅ a 2 . The exponent of a is 2, which is a non-negative integer. So, this term is a polynomial term.
The third term is 8 a , which can be written as 8 a 1/2 . The exponent of a is 2 1 , which is not a non-negative integer. Therefore, this term is not a polynomial term.
Conclusion Since the expression contains a term ( 8 a ) with a non-integer exponent, the entire expression is not a polynomial.
Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a ball, the growth of a population, or the behavior of electrical circuits. Understanding polynomials helps in making predictions and solving problems in these areas. For example, engineers use polynomials to design bridges and buildings, while economists use them to model economic growth.
The expression − a − 10 a 2 5 + 8 a is not a polynomial because it contains a term, 8 a , with a non-integer exponent ( 2 1 ). Polynomials must have non-negative integer exponents for all terms. Therefore, the answer is that it is not a polynomial.
;
