Fix the mistake: Monomials and Polynomials

Monomials and Polynomials ① What Are Monomials and Polynomials?

We will expand on the linear expressions learned in Grade 7 and work with various algebraic expressions. First, make sure you understand the difference between monomials and polynomials.

Monomials and Polynomials

An expression formed only by multiplying numbers and variables is called a monomial.
　Examples: 3x, −5a², 4xy, 7 (a number alone is also a monomial)

An expression represented as a sum of monomials is called a polynomial, and each individual monomial is called a term.
　Example: The terms of 3x + 2y − 5 are 3x, 2y, and −5.

Coefficients and Degrees

In a monomial, the part other than the variables is called the coefficient, and the number of variables being multiplied is called the degree.
　Example: The coefficient of −4x²y is −4, and its degree is 3 (x×x×y contains three variables).

The degree of a polynomial is the greatest of the degrees of its terms. A polynomial of degree n is called an nth-degree expression.
　Examples: 3x²−5x+2 is a second-degree expression, and 4ab+3a−1 is a second-degree expression.

Example ①
State the terms, the coefficient of each term, and the degree of the following expression.
　5x²y − 3xy + 7
Answer: The terms are 5x²y, −3xy, and 7. The coefficients are 5, −3, and 7, respectively. The degrees are 3 for 5x²y, 2 for −3xy, and 0 for 7. The degree of the entire polynomial is 3.

Like Terms

Terms whose variable parts are exactly the same (including their exponents) are called like terms. Like terms can be combined by adding or subtracting their coefficients.
　Example: 3x² and −5x² are like terms. 3x² and 3x are not like terms (their degrees are different).

Example ②
Combine the like terms in the following expression: 4x² + 3x − 7x² + 5x − 2
Answer: x² terms: 4x² − 7x² ＝ −3x². x terms: 3x + 5x ＝ 8x. Constant: −2.
Therefore, −3x² + 8x − 2.
Key Point
  • Monomial: product of numbers and variables. Polynomial: sum of monomials
  • Coefficient = part other than the variables; degree = number of variables being multiplied
  • Degree of a polynomial = greatest degree among its terms
  • Like terms have exactly identical variable parts, including exponents

Practice Problems

  1. State the coefficient and degree of each monomial.
    　(1) −6a²b　(2) 3xyz　(3) −x³
  2. What is the degree of the following polynomial: 2x²y − 3xy + 4x − 5
  3. Combine the like terms in the following expression: 6a − 2b + 3a + 5b − 7a

Answer and Explanation

  1. Answer:(1) Coefficient −6, degree 3　(2) Coefficient 3, degree 3　(3) Coefficient −1, degree 3
    Explanation: The degree is the number of variables. a²b is a×a×b, which contains three variables; xyz also contains three; x³ is x×x×x, which contains three. (3) The coefficient of −x³ is −1 (1 is omitted).
  2. Answer:Third-degree expression
    Explanation: The degrees of the terms are 3 for 2x²y, 2 for −3xy, 1 for 4x, and 0 for −5. The greatest is 3, so it is a third-degree expression.
  3. Answer:2a + 3b
    Explanation: a terms: 6a + 3a − 7a = 2a. b terms: −2b + 5b = 3b. Therefore, 2a + 3b.

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