Question

is each expression a polynomial? select polynomial or not a polynomial.

	polynomial	not a polynomial
$\frac{1}{2}y^3 + 5y^2$	$\bigcirc$	$\bigcirc$
$2j^{-5} - 5j^2 + 3j$	$\bigcirc$	$\bigcirc$

Explanation:

To determine if an expression is a polynomial, we use the definition: a polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, where variables have non - negative integer exponents.

For the expression \(2x\)

Step 1: Analyze the expression

The expression \(2x\) can be written as \(2x^{1}\). Here, the variable \(x\) has an exponent of \(1\), which is a non - negative integer. Also, it is a combination of a coefficient (\(2\)) and a variable (\(x\)) using multiplication (implicit in the form \(2x\)). So, \(2x\) is a polynomial.

For the expression \(\frac{1}{2}y^{3}+5y^{2}\)

Step 1: Analyze each term

• The first term \(\frac{1}{2}y^{3}\): The variable \(y\) has an exponent of \(3\) (a non - negative integer), and it is a product of a coefficient (\(\frac{1}{2}\)) and \(y^{3}\).
• The second term \(5y^{2}\): The variable \(y\) has an exponent of \(2\) (a non - negative integer), and it is a product of a coefficient (\(5\)) and \(y^{2}\).
• The two terms are combined using addition. Since all the exponents of the variable \(y\) are non - negative integers and the operations are addition and multiplication of coefficients and variables, \(\frac{1}{2}y^{3}+5y^{2}\) is a polynomial.

For the expression \(2j^{- 5}-5j^{2}+3j\)

Step 1: Analyze the first term

The first term is \(2j^{-5}\). We can rewrite \(j^{-5}\) as \(\frac{1}{j^{5}}\) using the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\) (\(a
eq0\), \(n>0\)). So, \(2j^{-5}=\frac{2}{j^{5}}\), which involves division by a variable. In a polynomial, variables cannot be in the denominator (because that would mean the exponent of the variable is negative, and polynomials require non - negative integer exponents for variables). So, \(2j^{-5}-5j^{2}+3j\) is not a polynomial.

Answer:

s:

• For \(2x\): polynomial
• For \(\frac{1}{2}y^{3}+5y^{2}\): polynomial
• For \(2j^{-5}-5j^{2}+3j\): not a polynomial