Algebraic expressions

Correct answer: B. \(18r-17\)

Explanation: Use the distributive property: \(7(2r-3)=14r-21\) and \(4(r+1)=4r+4\). Thus, \(14r-21+4r+4=18r-17\). Therefore, the correct expression is \(18r-17\). The option \(18r+25\) may result from incorrectly adding the constant terms with their signs. Exam tip: while expanding brackets, multiply the outside number by every term inside the bracket.

Correct answer: A. \(8a^2b-5ab\)

Explanation: \(5a^2b\) and \(3a^2b\) are like terms, so they add to \(8a^2b\). Similarly, \(2ab-7ab=-5ab\). Therefore, the simplified expression is \(8a^2b-5ab\). \(10a^3b\) is incorrect because exponents are not added when terms are added or subtracted. Exam tip: combine only terms with identical variable parts and powers.

Correct answer: C. 3

Explanation: Given x=2 and y=-1, x^2+xy+y^2=2^2+(2)(-1)+(-1)^2=4-2+1=3. Therefore, the correct answer is 3. The value 1 is only y^2; the terms x^2 and xy must also be included. Exam tip: the square of a negative number is positive, so (-1)^2=1.

Correct answer: C. -3x

Explanation: A linear term has a variable with exponent 1. In the given expression, 5x^2 has exponent 2, 2 is a constant term, and -3x has x raised to the power 1. Therefore, -3x is the linear term. In exams, identify the type of each term by checking the exponent of its variable.

Correct answer: A. \(a+2b\)

Explanation: On expanding the bracket, \(2(a+b)-a=2a+2b-a\). Combining like terms gives \(2a-a=a\), so the expression equals \(a+2b\). The option \(2a+b\) does not simplify the \(a\)-terms correctly. Exam tip: while expanding brackets, multiply the outside coefficient by every term inside the bracket.

Correct answer: C. 3

Explanation: The powers of the variable s in the expression are 3, 2, and 1, while the constant term −8 has power 0. Therefore, the highest power is 3, found in the term 4s^3. The number 8 is only part of the constant term, not a power. Exam tip: To find the degree of a polynomial, identify the greatest exponent of its variable.

Correct answer: C. 6

Explanation: In an algebraic term, the number that multiplies the variable part is called the numerical coefficient. In 6x^2y, 6 multiplies x^2y, so the numerical coefficient is 6. Here, x^2y is the variable part, while 2 is only the exponent of x. Exam tip: To identify a coefficient, separate the variables and their exponents from the term; the remaining number is the coefficient.

Correct answer: B. \(8x-17\)

Explanation: Using the distributive property, \(3(4x-5)=12x-15\) and \(-2(2x+1)=-4x-2\). Therefore, \(12x-15-4x-2=8x-17\). Hence, \(8x-17\) is correct. In \(8x-13\), the constant terms have been combined incorrectly. Exam tip: when a negative coefficient is outside a bracket, apply it to every term inside the bracket.

Correct answer: C. \(5(m-n)\)

Explanation: The difference between m and n is \(m-n\). Taking five times this entire difference gives \(5(m-n)\), so option C is correct. In \(5m-n\), only m is multiplied by 5, not the whole difference. Exam tip: When a phrase says “times the difference,” put the difference in brackets before multiplying.

Correct answer: A. \(6z^2-5z+3\)

Explanation: Combine like terms: \(10z^2-4z^2=6z^2\) and \(z-6z=-5z\). There is no other constant term to combine with \(3\), so it remains \(3\). Therefore, the simplified expression is \(6z^2-5z+3\). Option B incorrectly adds the \(z^2\) terms. Exam tip: add or subtract only terms having the same variable and exponent.

Correct answer: A. -12

Explanation: Given \(c=-2\), \(2c^3+c^2=2(-2)^3+(-2)^2=2(-8)+4=-16+4=-12\). Therefore, \(-12\) is correct. Since \(c^3\) has an odd exponent, it remains negative, whereas \(c^2\) has an even exponent and is positive. Exam tip: When substituting a negative value, check whether each exponent is odd or even.

Correct answer: C. Because powers of (a) and (b) are arranged differently

Explanation: Like terms are algebraic terms that have exactly the same variables with exactly the same exponents. Their coefficients may be different, but the variable pattern must match. In \(a^2b\), the exponent of a is 2 and the exponent of b is 1. In \(ab^2\), the exponent of a is 1 and the exponent of b is 2. The exponents are therefore interchanged, so the terms are not like terms. Option C states this idea.

The terms may look similar because both contain a and b, but merely having the same letters is not enough. For example, \(3a^2b\) and \(-5a^2b\) are like terms, while \(a^2b\) and \(ab^2\) are not. Options A, B and D do not describe the actual difference: the coefficients need not be the issue, variables are present, and these are not constants. Hence option C is correct.

Correct answer: A. \(5x-12\)

Explanation: On opening the bracket, \(-3(4-x)=-12+3x\), because \(-3\) multiplies both 4 and \(-x\). Therefore, \(2x-3(4-x)=2x-12+3x=5x-12\). In \(5x+12\), the sign of the constant term is incorrect. Exam tip: when multiplying a bracket by a negative number, check the sign of every term.

Correct answer: B. \(8rs-5r\)

Explanation: In \(8rs-5r\), the term \(8rs\) can be written as \(8\times rs\). Hence, the coefficient of \(rs\) is 8. In \(rs+8\), the coefficient of \(rs\) is 1 because \(rs=1\times rs\). Exam tip: To find a coefficient, identify the numerical factor multiplying the term.

Correct answer: A. \(-2x^2+8xy\)

Explanation: Here, \(2x^2\) and \(-4x^2\) are like terms, so \(2x^2-4x^2=-2x^2\). Similarly, \(3xy\) and \(5xy\) are like terms, giving \(3xy+5xy=8xy\). Therefore, the simplified expression is \(-2x^2+8xy\). The terms \(x^2\) and \(xy\) are not like terms, so they cannot be combined. Exam tip: add or subtract only terms with identical variables and powers.

Correct answer: B. 27

Explanation: Substituting the given values, (2x+y^2)=2(1)+5^2=2+25=27. Hence, the correct answer is 27. A result such as 12 may come from incorrectly treating y² as 2y. In exams, evaluate powers first, then multiplication and addition.

Correct answer: C. 5x+6

Explanation: Use the distributive property: \(6(x+1)-x=6x+6-x\). Combining the like terms \(6x\) and \(-x\) gives \(5x\), so the expression equals \(5x+6\). The option \(7x+6\) results from incorrectly treating \(6x-x\) as \(7x\). Exam tip: after opening brackets, add or subtract coefficients only of like terms.

Correct answer: A. (2x^2+2x+1)

Explanation: Simplifying a polynomial means combining only like terms. Like terms have the same variables with the same powers, so their coefficients can be added. In the expression \(x^2+x+x^2+x+1\), the two \(x^2\) terms combine to give \(2x^2\). The two x terms combine to give \(2x\). The number 1 is a constant and has no matching constant, so it remains 1. Therefore the simplified expression is \(2x^2+2x+1\), which is option A.

A useful check is to group the terms: \((x^2+x^2)+(x+x)+1\). Adding each group gives \(2x^2+2x+1\). We must not multiply the terms or change their powers; addition combines coefficients of identical variable parts. Thus option B incorrectly changes the coefficient and loses the x term, while C and D use incorrect powers. The grouping confirms option A.

Correct answer: B. 6

Explanation: Substituting \(n=-4\), we get \(n^2=(-4)^2=16\). Hence, \(\frac{n^2+n}{2}=\frac{16+(-4)}{2}=\frac{12}{2}=6\). Therefore, 6 is the correct answer. The distractor 8 results from wrongly ignoring the \(n=-4\) term in \(n^2+n\). Exam tip: always put a negative number in brackets before squaring it.

Correct answer: A. \(7a-b\)

Explanation: On expanding the brackets, \(3(a-b)+2(2a+b)=3a-3b+4a+2b\). Combining like terms gives \(3a+4a=7a\) and \(-3b+2b=-b\). Therefore, the simplified expression is \(7a-b\). The option \(7a+b\) results from combining the terms containing \(b\) incorrectly. Exam tip: multiply the number outside a bracket by every term inside it.

Correct answer: B. \(7x^2-5xy\)

Explanation: \(4x^2\) and \(3x^2\) are like terms, so they add to \(7x^2\). Similarly, \(-6xy\) and \(+xy\) are like terms, giving \(-6xy+xy=-5xy\). Hence, the simplified expression is \(7x^2-5xy\). \(12x^2-5xy\) is incorrect because \(4+3=7\), not 12. Exam tip: combine only terms with identical variable parts and exponents.

Correct answer: C. -1

Explanation: A constant term is a term that contains no variable. In \(2x^2+5x-1\), both \(2x^2\) and \(5x\) contain \(x\), whereas \(-1\) has no variable. Therefore, the constant term is \(-1\). The number \(5\) is the coefficient of \(5x\), not the constant term. Exam tip: identify the term with no letter or variable to find the constant term.

Correct answer: A. -4a - 30

Explanation: Answer: option A, −4a − 30. Simplify from the innermost brackets outward. First, 2(a−5)=2a−10. Therefore 4a−2(a−5) = 4a−(2a−10) = 4a−2a+10 = 2a+10. Substitute this into the original expression: 2a−3(2a+10). Now distribute −3 to both terms: −3(2a+10)=−6a−30. Combining like terms gives 2a−6a−30 = −4a−30. Thus A is correct. Option B has the wrong sign for the constant term. Options C and D have a positive coefficient of a because the negative multiplier was not distributed correctly. The common warning is that subtracting a bracket changes the signs of every term inside it.

Correct answer: B. 18 - 5x

Explanation: The governing concepts are the distributive law and the rule that a minus sign outside brackets changes the signs of every term inside. Begin with the innermost product: 2(4 − x) = 8 − 2x. The square-bracket expression is therefore 3x − (8 − 2x) = 3x − 8 + 2x = 5x − 8. Substituting into the complete expression gives 10 − (5x − 8). Apply the outer subtraction to both terms: 10 − 5x + 8 = 18 − 5x. Therefore option B is correct. Option A does not combine the x-terms correctly. Option C reverses the effect of the outer subtraction, and option D ignores the coefficient 5 formed from 3x + 2x.

Correct answer: A. 5x - 11

Explanation: The governing concepts are the distributive law, nested grouping symbols, and sign reversal when subtracting a bracketed expression. First simplify the braces: 4x − {3x + 1} = 4x − 3x − 1 = x − 1. Also expand the first product: 6(x − 2) = 6x − 12. The whole expression becomes (6x − 12) − (x − 1). Because the second bracket is subtracted, both of its signs change: 6x − 12 − x + 1 = 5x − 11. Thus option A is correct. Option B keeps the constant as −13 incorrectly. Options C and D use 7x, which results from adding x instead of subtracting it after removing the outer minus sign.