Algebraic expressions

\(x+7\) is an algebraic expression because it contains the variable \(x\), the constant 7, and an operation. \(x=7\) is an equation, while \(x+7>0\) is an inequality; they contain relation symbols and are not merely expressions. Exam tip: An expression generally does not contain relation symbols such as \(=, <, >\).

In (5x), the number multiplying (x) is (5). To identify a coefficient, look at the number attached to the variable.

A constant term is a term that contains no variable. In 3a+4, the term 3a contains the variable a, whereas 4 has no variable. Therefore, 4 is the constant term. The number 3 is the coefficient of 3a, not the constant term. Exam tip: Identify the term with no letter or variable to find the constant term.

The expression \(2m+6\) has two terms separated by the plus sign: \(2m\) and \(6\). Therefore, it has 2 terms. In \(2m\), 2 is the coefficient and \(m\) is the variable; they are not separate terms. Exam tip: Count only the parts separated by + or − signs.

A variable is a letter or symbol whose value can change. In (6y-2), the value of y can vary, so y is the variable. Here, 6 is the coefficient of y and 2 is a constant; 8 does not occur in the expression. Exam tip: identify letters as variables and standalone numbers as constants.

Given x = 2, substitute 2 for x in the expression x + 5. This gives 2 + 5 = 7, so the correct answer is 7. Option 5 is only the constant term; the value of x must also be added. Exam tip: To evaluate an expression, first replace each variable with its given value and then simplify.

The expression \(x^2+3x-5\) has three terms: \(x^2\), \(3x\), and \(-5\), so it is a trinomial. \(4a-7\) is a binomial, \(9y\) is a monomial, and \(p+q+r+s\) has four terms. Exam tip: count the parts separated by + or − signs to find the number of terms.

In \(9-r\), the minus sign separates the terms. The two terms are \(9\) and \(-r\), so the correct answer is 2. It is not one term because its two parts are separated by a minus sign. Exam tip: Count each part separated by a \(+\) or \(-\) sign as a term.

4x and 3x are like terms because both contain x to the power 1. Therefore, add their coefficients: \(4x+3x=(4+3)x=7x\). The option 7 is only the sum of the coefficients and incorrectly leaves out x. Exam tip: Only terms with the same variable and the same exponent can be added or subtracted.

\(8b\) and \(5b\) are like terms because both contain \(b\) to the first power. Subtract their coefficients: \(8b-5b=(8-5)b=3b\). \(13b\) would result from addition, while \(3\) incorrectly omits the variable \(b\). Exam tip: add or subtract only the coefficients of like terms.

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.

Answer: option A, 6a. A constant term is a term that contains no variable. The expression 6a has only one visible term, and that term contains the variable a. Therefore there is no separate number-only term; in polynomial form we may write it as 6a + 0, so its constant term is understood to be 0. Do not mistake the number 6 for the constant term: 6 is the coefficient of a because it multiplies the variable. In 9−x, the constant term is 9. In p+4, it is 4. In 7y−2, it is −2, including its negative sign. Memory cue: coefficient sits beside a variable; constant term stands alone.

Answer: option C, 1. A coefficient is the numerical factor multiplied by a variable term. In x² + 2x + 1, the first term is x². No number is visibly written before x², but algebraic convention says that an unwritten coefficient is 1: x² means 1×x². Therefore the coefficient of x² is 1. Option A, 2, is the coefficient of x, because the second term is 2x. Option B is incorrect because the term x² is present, not absent. Option D is the variable itself, not a numerical coefficient. The final 1 is the constant term. Memory cue: when a term begins directly with a variable, its hidden coefficient is 1.

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.