Question

determine if the expression $b^2a^2 - c^3$ is a polynomial or not. if it is a polynomial, state the type and degree of the polynomial. answer attempt 1 out of 3 the given expression a polynomial.

Explanation:

Step1: 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. In the expression \(b^{2}a^{2}-c^{3}\), we have variables \(a\), \(b\), and \(c\) with non - negative integer exponents (the exponent of \(a\) is 2, the exponent of \(b\) is 2, and the exponent of \(c\) is 3) and the operations of multiplication (\(b^{2}a^{2}\) is a product of \(b^{2}\) and \(a^{2}\)) and subtraction. So, the expression is a polynomial.

Step2: Determine the type of the polynomial

The number of terms in a polynomial determines its type. A monomial has 1 term, a binomial has 2 terms, and a trinomial has 3 terms. The expression \(b^{2}a^{2}-c^{3}\) can be written as \(b^{2}a^{2}+(-c^{3})\), which has two terms. So, it is a binomial.

Step3: Determine the degree of the polynomial

The degree of a term in a polynomial is the sum of the exponents of its variables. For the term \(b^{2}a^{2}\), the sum of exponents is \(2 + 2=4\). For the term \(-c^{3}\), the sum of exponents is 3. The degree of the polynomial is the highest degree among its terms. Since 4>3, the degree of the polynomial is 4.

Answer:

The given expression \(\boldsymbol{is}\) a polynomial. It is a binomial (type) with degree \(\boldsymbol{4}\).