Algebraic expressions

Correct answer: A. \(x+7\)

Explanation: \(x+7\) 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 \(=, <, >\).

Correct answer: B. (5)

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

Correct answer: C. 4

Explanation: 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.

Correct answer: D. 2

Explanation: 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.

Correct answer: D. y

Explanation: 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.

Correct answer: A. 7

Explanation: 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.

Correct answer: A. \(x^2+3x-5\)

Explanation: 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.

Correct answer: C. 2

Explanation: 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.

Correct answer: D. 7x

Explanation: 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.

Correct answer: A. 3b

Explanation: \(8b\) 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.

Correct answer: C. 3

Explanation: 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.

Correct answer: A. Constant expression

Explanation: 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.

Correct answer: B. 12

Explanation: 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.

Correct answer: A. 6a

Explanation: 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.

Correct answer: C. 1

Explanation: 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.

Correct answer: D. -1

Explanation: 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.

Correct answer: A. 3

Explanation: 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.

Correct answer: A. The expression has three terms: \(5a^2\), \(-3a\), and \(1\).

Explanation: 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.

Correct answer: C. \(3a\)

Explanation: 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 \(1\).

Correct answer: A. \(7x+2\)

Explanation: \(4x\) 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.

Correct answer: C. Term

Explanation: 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.

Correct answer: A. \(2x+3\)

Explanation: 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.

Correct answer: B. 2

Explanation: 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.

Correct answer: C. 2

Explanation: 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.

Correct answer: B. 2

Explanation: 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.