Identify terms and coefficients in polynomials

A polynomial is understood as a sum or difference of terms, where each term is a numerical coefficient multiplied by a variable raised to a nonnegative integer exponent; the coefficient includes its sign, and a missing coefficient is understood as $1$ or $-1$ . Learners distinguish terms, coefficients, variable parts, exponents, and constant terms, recognizing like terms as having identical variable parts; the focus is one-variable expressions with numerical coefficients, not multivariable polynomials or abstract coefficient systems.

Detailed Explanation: Identify terms and coefficients in polynomials

A polynomial is made of terms joined by addition or subtraction. To identify each part:

Separate the terms at the plus or minus signs.
For each term, identify:
- the coefficient: the number multiplying the variable, including its sign;
- the variable part: the letter and its exponent;
- the exponent: the power on the variable.
A term with no variable is a constant term.
Terms with the same variable part are like terms.

Consider the polynomial

3x^2-x+6+2x^2.

Step 1: Identify the terms

The terms are

3x^2,\qquad -x,\qquad 6,\qquad 2x^2.

The minus sign belongs to the term after it, so the second term is $(-x)$ , not just $x$ .

Step 2: Identify each part

Term	Coefficient	Variable part	Exponent
$(3x^2)$	$3$	$(x^2)$	$2$
$(-x)$	$(-1)$	$x$	$1$
$6$	$6$	none	none
$(2x^2)$	$2$	$(x^2)$	$2$

The coefficient of $(-x)$ is $(-1)$ because a missing coefficient is understood to be $1$ , and the negative sign makes it $(-1)$ . Also, $x$ means $(x^1)$ .

The term $6$ is a constant term because it has no variable.

Finally, $(3x^2)$ and $(2x^2)$ are like terms because they have the same variable part, $(x^2)$ . The terms $(-x)$ and $6$ are not like terms with them.

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Identify Coefficient - Polynomial and Coefficient to Term

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