Question

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. 15a2b3+12a3b8−$$13b5+12b615a^2b^3$+12a^3b^8$-13b^5$+12b^6$$

Accepted Answer

Step 1: Understand the definition of a polynomial. A polynomial is an algebraic expression consisting of terms that are variables raised to non-negative integer exponents, multiplied by coefficients, and combined using addition or subtraction. Step 2: Examine each term in the expression $$15a^{2}b$$^{3}$$ + 12a$$^{3}$$b^{8}$$ - $$13b^{5}$$ + $$12b^{6}. $$Check that all variables have whole number exponents and that there are no variables in denominators or under roots. Step 3: Since all terms have variables with non-negative integer exponents and are combined by addition and subtraction, confirm that the entire expression is a polynomial. Step 4: Count the number of terms in the polynomial. Here, there are four terms: $$15a^{2}b$$^{3}$$, 12a$$^{3}$$b^{8}$$, $$-13b^{5}$$, and $$12b^{6}. $$Since there are four terms, it is neither a monomial (1 term), binomial (2 terms), nor trinomial (3 terms). Step 5: Determine the degree of the polynomial by finding the term with the highest sum of exponents. Calculate the degree of each term by adding the exponents of the variables in that term, then identify the largest sum as the degree of the polynomial.

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. $15 a^{2} b^{3} + 12 a^{3} b^{8} - 13 b^{5} + 12 b^{6}$

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Key Concepts

Definition of a Polynomial

Degree of a Polynomial

Classification by Number of Terms

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. 15a2b3+12a3b8−13b5+12b615a^2b^3+12a^3b^8-13b^5+12b^6