Question
Parts of algebraic expressions
Answers
GPT-5 mini
Here are the common parts of algebraic expressions, with short definitions and examples.
Basic pieces
- Variable: a symbol that stands for a number (usually x, y, a).
Example: in 5x, x is the variable.
- Constant: a fixed number (no variable).
Example: 7 in 3x + 7.
- Coefficient: the number multiplying a variable or product of variables.
Example: in −4x^2y, the coefficient is −4. If no number is written (x or −x), the coefficient is 1 or −1.
- Exponent (power): the small raised number showing repeated multiplication of a base.
Example: in x^3, 3 is the exponent and x is the base.
- Term: a single part of an expression separated by + or − signs (a product of constants and variables with exponents).
Example: 3x^2y, −4xy, and 7 are the terms of 3x^2y − 4xy + 7.
- Factor: a quantity multiplied with others inside a term. Terms can have factors (numbers, variables, expressions).
Example: in 6ab, the factors are 6, a, and b.
- Like terms: terms that have the same variable part (same variables with same exponents). Coefficients can differ.
Example: 5x^2y and −3x^2y are like terms; 2x^2 and 2x are not.
- Degree:
- Degree of a term (total degree): sum of exponents of its variables.
Example: degree of 3x^2y is 2 + 1 = 3.
- Degree of a polynomial: the highest degree among its terms.
Example: degree of 4x^3 − x + 2 is 3.
- Leading term / leading coefficient: in a polynomial written in standard form (descending powers), the first (highest-degree) term and its coefficient.
Example: in 5x^4 − 2x^2 + x, the leading term is 5x^4 and the leading coefficient is 5.
- Constant term: the term with no variables.
Example: 7 is the constant term in x^2 + 3x + 7.
Types of expressions
- Monomial: one term (e.g., 6x^2y).
- Binomial: two terms (e.g., x + 5).
- Trinomial: three terms (e.g., x^2 + x + 1).
- Polynomial: a sum of one or more terms.
Quick examples — identify parts
- Expression: 3x^2y − 4xy + 7
- Terms: 3x^2y, −4xy, 7
- Coefficients: 3, −4, (7 is constant)
- Variable parts: x^2y, xy, (none)
- Degree of terms: 3, 2, 0
- Degree of polynomial: 3
- Constant term: 7
- Expression: −x^3 + 2x − 5
- Leading term: −x^3
- Leading coefficient: −1
- It is a polynomial of degree 3.
Combining like terms (simple use)
- 3x^2y − 4xy + 2x^2y = (3 + 2)x^2y − 4xy = 5x^2y − 4xy
If you want, I can: give more examples, make practice problems, or show how these parts are used when factoring or solving. Which would you like?
Basic pieces
- Variable: a symbol that stands for a number (usually x, y, a).
Example: in 5x, x is the variable.
- Constant: a fixed number (no variable).
Example: 7 in 3x + 7.
- Coefficient: the number multiplying a variable or product of variables.
Example: in −4x^2y, the coefficient is −4. If no number is written (x or −x), the coefficient is 1 or −1.
- Exponent (power): the small raised number showing repeated multiplication of a base.
Example: in x^3, 3 is the exponent and x is the base.
- Term: a single part of an expression separated by + or − signs (a product of constants and variables with exponents).
Example: 3x^2y, −4xy, and 7 are the terms of 3x^2y − 4xy + 7.
- Factor: a quantity multiplied with others inside a term. Terms can have factors (numbers, variables, expressions).
Example: in 6ab, the factors are 6, a, and b.
- Like terms: terms that have the same variable part (same variables with same exponents). Coefficients can differ.
Example: 5x^2y and −3x^2y are like terms; 2x^2 and 2x are not.
- Degree:
- Degree of a term (total degree): sum of exponents of its variables.
Example: degree of 3x^2y is 2 + 1 = 3.
- Degree of a polynomial: the highest degree among its terms.
Example: degree of 4x^3 − x + 2 is 3.
- Leading term / leading coefficient: in a polynomial written in standard form (descending powers), the first (highest-degree) term and its coefficient.
Example: in 5x^4 − 2x^2 + x, the leading term is 5x^4 and the leading coefficient is 5.
- Constant term: the term with no variables.
Example: 7 is the constant term in x^2 + 3x + 7.
Types of expressions
- Monomial: one term (e.g., 6x^2y).
- Binomial: two terms (e.g., x + 5).
- Trinomial: three terms (e.g., x^2 + x + 1).
- Polynomial: a sum of one or more terms.
Quick examples — identify parts
- Expression: 3x^2y − 4xy + 7
- Terms: 3x^2y, −4xy, 7
- Coefficients: 3, −4, (7 is constant)
- Variable parts: x^2y, xy, (none)
- Degree of terms: 3, 2, 0
- Degree of polynomial: 3
- Constant term: 7
- Expression: −x^3 + 2x − 5
- Leading term: −x^3
- Leading coefficient: −1
- It is a polynomial of degree 3.
Combining like terms (simple use)
- 3x^2y − 4xy + 2x^2y = (3 + 2)x^2y − 4xy = 5x^2y − 4xy
If you want, I can: give more examples, make practice problems, or show how these parts are used when factoring or solving. Which would you like?