Algebraic expressions

Parts of an expression separated by plus (+) or minus (−) signs are called terms. Here, the terms are \(2x\), \(3y\), and \(5\). Therefore, there are 3 terms. Counting 2 because there are two variables would be incorrect; terms are counted as separate algebraic parts. Exam tip: Split an expression at + and − signs to count its terms.

The governing concept is classifying an algebraic expression by its terms and variables. The number 7 has no variable and does not change its value, so it is a constant expression; it may also be described as a monomial with a constant term. Therefore option A is correct. A binomial must contain two unlike terms, and a trinomial must contain three terms, so neither B nor C applies. “Unlike term” is not the appropriate type here: unlike terms are compared with other terms and differ in variable part or powers. A number by itself is still a valid algebraic expression.

The term 12pq can be written as 12 × p × q. Here, 12 is the numerical factor multiplying the variables p and q, so the numerical coefficient is 12. The expression pq is the variable part, not the coefficient. Exam tip: The number that multiplies the variables in a term is its numerical coefficient.

The governing concept is that a constant term is the term containing no variable. In the expression 6a, the only term contains the variable a, so there is no separately written number-only term. Therefore its constant term is 0. The other displayed expressions are quotients that simplify to 1 wherever their denominators are non-zero; they are not intended as single polynomial terms in this question and each visibly contains a constant within the brackets. Option C is therefore the intended and unambiguous answer. The coefficient 6 is not the constant term because it multiplies a.

The governing concept is that the coefficient is the numerical factor multiplying a term. In x² + 2x + 1, the term x² has no number visibly written before it. In algebra, an unwritten numerical factor is understood to be 1, so x² means 1 × x². The number 2 is the coefficient of x, not of x²; 1 is the constant term; and x is a variable expression rather than the numerical coefficient. Hence option C is correct. Reading each term separately prevents confusing the coefficient of one term with the coefficient of another.

The term \(-y\) can be written as \(-1\times y\). Therefore, the coefficient of \(y\) is \(-1\). Option 1 would be the coefficient of \(y\), but the negative sign makes the coefficient negative here. Exam tip: The number multiplying a variable, including its sign, is its coefficient.

Given p = 4, substitute 4 for p in p - 1: 4 - 1 = 3. Therefore, the correct answer is 3. Option 4 is the value of p itself, not the value of p - 1. Exam tip: First substitute the given value of the variable, then simplify the expression.

A trinomial is an algebraic expression with exactly three terms. The terms of \(5a^2-3a+1\) are \(5a^2\), \(-3a\), and \(1\), so it is a trinomial. The highest power being 2, as stated in option B, makes it a quadratic expression; it does not explain why it is a trinomial. Exam tip: count the terms separated by plus or minus signs.

The term \(a\) is added three times: \(a+a+a=3\times a=3a\). Therefore, \(3a\) is correct. \(a^3\) means \(a\times a\times a\), so it represents repeated multiplication, not repeated addition. Exam tip: When adding like terms, add their coefficients; the coefficient of \(a\) is \(1\).

\(4x\) and \(3x\) are like terms, so adding their coefficients gives \(4x+3x=7x\). The constant term \(2\) has no like term, so it remains unchanged. Therefore, the simplified expression is \(7x+2\). Choosing \(7x\) would incorrectly omit the constant term \(2\). Exam tip: Add or subtract only terms with the same variable and the same power.

In the expression
(2x+5), the parts separated by the plus sign are 2x and 5. Therefore, 2x is a term. Here, x is the variable and 5 is the constant; the complete expression (2x+5) is not an equation because it has no equal sign. Exam tip: Identify terms in an expression by locating the + or − signs.

In \(2x+3\), \(x\) is the only variable; 2 and 3 are constants. \(a+b\) has two variables, \(a\) and \(b\), while \(pq+1\) has \(p\) and \(q\). \(7\) is a constant expression and has no variable. Exam tip: count distinct letters as variables, not coefficients or constants.

In the expression
(3u+4v), u and v are letters whose values can vary, so they are variables. The numbers 3 and 4 are coefficients, not variables. Therefore, there are 2 variables. Exam tip: Count the distinct letters whose values can change in an expression.

In \(x^2\), 2 is the power (exponent) of \(x\). It means \(x^2=x\times x\). Option 1 is not the exponent here; it would be the coefficient of \(x\) in the expression \(x\). Exam tip: The number written as a superscript on a variable gives its power.

In an algebraic term, the numerical factor is called the coefficient and the exponent attached to a variable gives its power. In 5x², 5 is the coefficient and the superscript 2 is the power of x. Therefore option B is correct. Option A identifies the coefficient, option D is the implicit power of x when no exponent is written, and 7 is obtained by an irrelevant addition.

The terms are 3x^2, 2x, and 5. The powers of x are 2 and 1 respectively, while the constant term 5 has power 0. Hence, the highest power is 2. Note that 3 is the coefficient of x^2, not its power. Exam tip: To find the highest power of a polynomial, look at the exponent of the variable, not the coefficient.

Option D, 11, is a constant because its value is fixed and it contains no variable. The expressions m, 2m, and m+2 contain the variable m, so their values can change when m changes. Exam tip: A term with no letter or variable is a constant.

A constant term is a term that contains no variable. In \((10x+3y-4)\), \(10x\) and \(3y\) contain variables, whereas \(-4\) has no variable. Therefore, the constant term is \(-4\), not just \(4\), because the negative sign is part of the term. Exam tip: always retain the sign while identifying a constant term.

6x and 2x are like terms, so their coefficients are added: 6x + 2x = 8x. The constant 1 has no like x-term, so it remains separate. Therefore, the simplified expression is 8x + 1. Option 8x incorrectly omits the constant term 1. Exam tip: Add or subtract only terms with the same variable and exponent.

\(9a\) and \(-3a\) are like terms, so their coefficients combine as \(9-3=6\). Hence, \(9a+2b-3a=6a+2b\). The term \(2b\) cannot be combined with the terms containing \(a\), because they have different variables. Exam tip: Add or subtract only terms with the same variables raised to the same powers.