Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial, or trinomial.
1. 7a^2b + 3b^2 – a^2b
2. 6g^2h^3k
To determine whether each expression is a polynomial, we need to understand what a polynomial is.
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication, but not division or square roots. The variables and exponents must be whole numbers (non-negative integers).
Let's analyze each expression:
1. 7a^2b + 3b^2 – a^2b
In this expression, we have three terms: 7a^2b, 3b^2, and -a^2b.
Each term consists of a coefficient (the number in front of the variables) multiplied by variables raised to a power (in this case, exponents of a and b).
Since all the terms are combined using addition and subtraction, and the variables and exponents are whole numbers, we can conclude that this expression is a polynomial.
To find the degree of the polynomial, we look at the term with the highest total exponent. In this case, the highest exponent is 2 (from a^2b). Therefore, the degree of the polynomial is 2.
Regarding the classification as monomial, binomial, or trinomial, we count the number of terms:
- Monomial: A polynomial with only one term.
- Binomial: A polynomial with exactly two terms.
- Trinomial: A polynomial with exactly three terms.
In this case, the expression has three terms, so it is classified as a trinomial.
2. 6g^2h^3k
In this expression, we have only one term: 6g^2h^3k.
Since it contains only one term and satisfies the conditions for a polynomial (variables and exponents are whole numbers, combined using multiplication), we can say that this is a polynomial.
The highest exponent in this expression is 3 (from h^3), so the degree of the polynomial is 3.
Since it has only one term, it is classified as a monomial.
To summarize:
1. 7a^2b + 3b^2 – a^2b is a polynomial of degree 2 and a trinomial.
2. 6g^2h^3k is a polynomial of degree 3 and a monomial.
To determine whether each expression is a polynomial, we need to check if the terms satisfy two conditions:
1. Each term consists of variables raised to non-negative whole number exponents.
2. The coefficients of each term are real numbers.
Let's analyze each expression:
1. 7a^2b + 3b^2 - a^2b
This expression satisfies both conditions. Each term consists of variables raised to non-negative whole number exponents (a^2b and b^2), and the coefficients for each term (7, 3, and -1) are real numbers.
To find the degree, we add up the exponents of the variables in each term. The highest degree within the expression is 2 (from the term 7a^2b).
Since there are three terms, this expression is a trinomial.
2. 6g^2h^3k
This expression also satisfies both conditions. Each term consists of variables raised to non-negative whole number exponents (g^2h^3k), and the coefficient (6) is a real number.
To find the degree, we add up the exponents of the variables in the term. The highest degree within the expression is 6 (from the term 6g^2h^3k).
Since there is only one term, this expression is a monomial.
To summarize:
1. 7a^2b + 3b^2 - a^2b is a trinomial of degree 2.
2. 6g^2h^3k is a monomial of degree 6.
if there are all positive exponents, then it is a polynomial.
Terms are separated by +/- signs, so just count the terms to determine the type.
Add the exponents in each term; the highest sum is the degree of the polynomial.