Verified Answer :

  • Define a monomial as a single-term expression and a binomial as a two-term expression.
  • Simplify like terms first, such as in \(wr + 5wr = 6wr\).
  • Classify the expressions:
  • a. \(3y\): Monomial
  • b. \(-9 + h\): Binomial
  • c. \(wr + 5wr\): Monomial
  • d. \(-7agh\): Monomial
  • e. \(2\): Monomial
  • f. \(k + 12\): Binomial
  • Final classifications: \(\boxed{\text{a: Monomial, b: Binomial, c: Monomial, d: Monomial, e: Monomial, f: Binomial}}\).

Explanation

  1. Defining Monomials and Binomials

A monomial is a polynomial consisting of exactly one term. A term is a product of numbers and variables, and constants are also considered monomials. A binomial is a polynomial consisting of exactly two non-like terms, meaning they cannot be combined further through addition or subtraction. To classify each given expression, we first check if it can be simplified by combining like terms. If it simplifies to one term, it is a monomial; if it results in two distinct non-like terms, it is a binomial.

  1. Evaluating Each Expression

Now we analyze each expression step-by-step:

A\(3y\): This expression consists of a single term (a product of a constant and a variable). It cannot be simplified further. Therefore, it is a monomial.
B\((-9) + h\): This expression consists of two distinct, non-like terms: a constant \(-9\) and a variable \(h\). They cannot be added together. Therefore, it is a binomial.
C\(wr + 5wr\): This expression contains two terms with the same variable part, \(wr\). We can combine these like terms: \(1wr + 5wr = 6wr\). After simplification, we have only one term. Therefore, it is a monomial.
D\(-7agh\): This expression is a product of a constant and three variables. It consists of only one term. Therefore, it is a monomial.
E\(2\): A constant is a polynomial of degree zero. It consists of a single term. Therefore, it is a monomial.

f. \(k + 12\): This expression consists of two distinct, non-like terms: a variable \(k\) and a constant \(12\). They cannot be added together. Therefore, it is a binomial.

  1. Conclusion

In summary, we have classified the expressions as follows:

A\(3y\) is a monomial.
B\((-9) + h\) is a binomial.
C\(wr + 5wr\) is a monomial.
D\(-7agh\) is a monomial.
E\(2\) is a monomial.

f. \(k + 12\) is a binomial.

Examples

In algebra, understanding the difference between monomials and binomials is crucial for performing operations like addition, subtraction, and multiplication of polynomials. For example, when calculating the area of a rectangle with length \((x + 3)\) and width \(5\), you are multiplying a binomial \((x + 3)\) by a monomial \((5)\). Distinguishing these structures helps you apply the distributive property correctly, leading to a simplified polynomial: \(5(x + 3) = 5x + 15\). Identifying terms correctly prevents errors in combining values, ensuring accuracy in algebraic simplification.

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