Algebraic expressions

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

Explanation: Add the like terms in the two expressions: 2x^2+x^2=3x^2, -5x+4x=-x, and 3+(-8)=-5. Therefore, the sum is (3x^2-x-5). Option B incorrectly adds the x-terms. Exam tip: while adding polynomials, combine only terms with the same variable and exponent.

Correct answer: C. \(6x^2-6x+6\)

Explanation: While subtracting the second polynomial, change the sign of each of its terms: \((9x^2-4x+1)-(3x^2+2x-5)=9x^2-4x+1-3x^2-2x+5\). Combining like terms gives \(6x^2-6x+6\). Option A has an incorrect linear term because \(-4x-2x=-6x\). Exam tip: change every sign inside the polynomial being subtracted before removing the brackets.

Correct answer: B. They are unlike terms because powers are different

Explanation: Two algebraic terms are like terms only when the exponent of every corresponding variable is the same. In \(x^2y\), x has exponent 2 and y has exponent 1. In \(xy^2\), x has exponent 1 and y has exponent 2. Although both terms contain the same two variables, their powers do not match. Therefore they are unlike terms, so option B is correct.

The order of writing the variables is not the real issue; the exponents attached to each variable must be compared. For instance, \(4x^2y\) and \(-7x^2y\) would be like terms because both have the pattern \(x^2y\). But \(x^2y\) and \(xy^2\) have different variable parts and cannot be combined by ordinary addition. They are not constants, and multiplying their variable parts is not what the question asks. Hence B is the correct statement.

Correct answer: A. -28

Explanation: Given \(a=1\) and \(b=-2\), we have \(a^2=1\), \(b^2=4\), and \(b^3=-8\). Thus, \(4a^2b-3ab^2+b^3=4(1)(-2)-3(1)(4)+(-8)=-8-12-8=-28\). Therefore, \(-28\) is correct. The option \(-12\) is only the value of the second term, not of the complete expression. Exam tip: an odd power of a negative number is negative, while an even power is positive.

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

Explanation: On expanding the brackets, \(2(4x-3y)=8x-6y\) and \(-5(x-y)=-5x+5y\). Thus, the expression becomes \(8x-6y-5x+5y+3y\). Combining like terms gives \((8x-5x)+(-6y+5y+3y)=3x+2y\). The distractor \(3x-8y\) results from incorrectly handling the sign of the \(y\)-term while expanding \(-5(x-y)\). Exam tip: When a negative coefficient multiplies a bracket, apply it to every term inside the bracket.

Correct answer: A. \(7\)

Explanation: In \(x^2+kx+16\), the term containing \(x\) is \(kx\). The coefficient of \(x\) in this term is \(k\). Since the coefficient of \(x\) is given as \(12\), \(k=12\). Therefore, \(k-5=12-5=7\). Option \(12\) is the value of \(k\), not of \(k-5\). Exam tip: identify the factor multiplying the variable to find its coefficient.

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

Explanation: The square of \(x\) is \(x^2\), so three times its square is \(3x^2\). Twice \(y\) is \(2y\). The phrase “the difference between A and B” means \(A-B\), so the required expression is \(3x^2-2y\). Option C reverses the order of subtraction and represents \(2y-3x^2\). Exam tip: Write the square first, then apply the numerical coefficient.

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

Explanation: Combine like terms: \(4x^2-x^2=3x^2\), \(-3xy+5xy=2xy\), and \(2y^2-7y^2=-5y^2\). Hence, the simplified expression is \(3x^2+2xy-5y^2\). Option B uses an incorrect coefficient for \(x^2\). Exam tip: add or subtract only terms having the same variables with the same powers.

Correct answer: B. 24

Explanation: Substitute the value directly into the expression:

\(2x(x-5)+4x^2=2\times3\times(3-5)+4\times3^2\)

\(=6\times(-2)+4\times9=-12+36=24\). Therefore, the correct answer is 24. A value such as 30 may result from an error while calculating the negative term \(-12\). In exams, evaluate powers such as \(x^2\) first and check signs inside brackets carefully.

Correct answer: B. -2

Explanation: Only like terms can be added or subtracted. The \(x^2y\) terms are \(3x^2y\) and \(-5x^2y\), so their sum is \((3-5)x^2y=-2x^2y\). Therefore, the coefficient of \(x^2y\) is \(-2\). The terms \(2xy^2\) and \(8xy^2\) are not like terms of \(x^2y\), so they do not affect this coefficient. Exam tip: Before combining terms, check that both the variables and their powers are identical.

Correct answer: A. \(x(4x-9)\)

Explanation: Using the distributive law, \(x(4x-9)=x\cdot4x-x\cdot9=4x^2-9x\). Therefore, option A is correct. Option B expands to \(4x^2-36x\), so it is not equivalent even though it looks similar. Exam tip: To check a factored expression, multiply each term outside the bracket by every term inside it.

Correct answer: C. 10

Explanation: Adding \(p+q=6\) and \(p-q=2\) gives \(2p=8\), so \(p=4\). Substituting \(p=4\) into \(p-q=2\) gives \(q=2\). Hence, \(3p-q=3(4)-2=10\). The value \(12\) is only \(3p\); \(q\) must also be subtracted. Exam tip: Add or subtract a pair of linear equations first to eliminate one variable.

Correct answer: A. \(3a^2-7ab+4b^2\)

Explanation: To subtract the second polynomial, change the sign of each of its terms: \((5a^2-2ab+3b^2)-(2a^2+5ab-b^2)=5a^2-2ab+3b^2-2a^2-5ab+b^2\). Combining like terms gives \(3a^2-7ab+4b^2\). In option B, the sign of the \(ab\) term is incorrect. Exam tip: when a minus sign comes before brackets, reverse the signs of every term inside the brackets.

Correct answer: C. 3

Explanation: The terms of 2x² + 5x + 3 are 2x², 5x, and 3. Here, 3 is the constant term because it contains no x. Removing 3 leaves 2x² + 5x, which has exactly two terms and no constant term. If 5x were removed, the constant term 3 would still remain. Exam tip: A term with no variable is called a constant term.

Correct answer: C. 8

Explanation: Substituting the given values, \(x^2y^2=(-1)^2\times2^2=1\times4=4\), while \(2xy=2\times(-1)\times2=-4\). Hence, \(x^2y^2-2xy=4-(-4)=8\). The option 4 results from missing the rule for subtracting a negative number. Exam tip: when a negative term is subtracted, change it to addition.

Correct answer: A. \(5x+6y\)

Explanation: On expanding, \(2(3x-y)=6x-2y\) and \(4(x+2y)=4x+8y\). Therefore, \(6x-2y+4x+8y-5x=(6+4-5)x+(-2+8)y=5x+6y\). In option B, the sign of the \(y\)-term is incorrect because \(-2y+8y=+6y\). Exam tip: after removing brackets, combine terms with the same variable separately.

Correct answer: C. \(3\)

Explanation: The total degree of the monomial \(x^3y^2\) is the sum of the exponents of all its variables: \(3+2=5\). The power of \(y\) is \(2\). Therefore, the required difference is \(5-2=3\). Note that \(5\) is the total degree, not the difference. Exam tip: To find the total degree of a monomial, add the exponents of every variable in it.

Correct answer: A. 56

Explanation: Given \(x^2+x=20\). Taking 3 common from the first two terms, \(3x^2+3x-4=3(x^2+x)-4\). Substituting the given value gives \(3(20)-4=60-4=56\). Therefore, 56 is correct. Option 60 is a close distractor because it results from forgetting to subtract 4. Exam tip: Rewrite the expression to contain the exact group given in the question before substituting.

Correct answer: D. \(5xy\)

Explanation: Combining like terms gives \(7x^2y+4x^2y=11x^2y\) and \(-3xy+2xy=-xy\). Therefore, the simplified expression is \(11x^2y-xy-6\), which does not contain the term \(5xy\). The close distractor \(-xy\) does remain because the coefficients of the \(xy\) terms add to \(-3+2=-1\), not \(5\). Exam tip: Add or subtract only those terms that have exactly the same variables raised to the same powers.

Correct answer: C. Both are equal

Explanation: Taking the common factor \(xy\) from the first expression gives \(x^2y+xy^2=xy(x+y)\). Therefore, the two expressions are equal for every value of \(x\) and \(y\). Substituting the given values, \(2^2\times1+2\times1^2=6\) and \(2\times1\times(2+1)=6\). Hence, neither expression is greater. Exam tip: factor out the common term \(xy\) to check such equivalence quickly.

Correct answer: A. \(5x^2+3xy-4y^2\)

Explanation: Combine like terms: \(8x^2-3x^2=5x^2\), \(-6xy+9xy=3xy\), and \(y^2-5y^2=-4y^2\). Hence, the simplified expression is \(5x^2+3xy-4y^2\). In option B, the coefficients of the \(x^2\) and \(y^2\) terms have been combined incorrectly. Exam tip: add or subtract only terms with the same variables raised to the same powers.

Correct answer: A. (2x^2-2x+9)

Explanation: When a whole polynomial is subtracted, the minus sign must be distributed to every term inside its brackets. Starting with \(4x^2-7x+6-(2x^2-5x-3)\), change the signs in the second bracket: \(-(2x^2-5x-3)=-2x^2+5x+3\). The expression becomes \(4x^2-7x+6-2x^2+5x+3\).

Now combine like terms. The quadratic terms give \(4x^2-2x^2=2x^2\), the linear terms give \(-7x+5x=-2x\), and the constants give \(6+3=9\). Hence the simplified expression is \(2x^2-2x+9\), which is option A. Options B and C result from failing to combine or change signs correctly, while option D has the wrong sign for the linear term. The supplied answer and explanation are correct.

Correct answer: B. (-7)

Explanation: Direct answer: Option B, \\(-7\\). Substitute \\(x=-2\\) carefully into every occurrence of x: \\(x^4+2x^3-3x^2+5=(-2)^4+2(-2)^3-3(-2)^2+5\\). Now calculate the powers: \\((-2)^4=16\\) because an even power is positive; \\((-2)^3=-8\\) because an odd power keeps the negative sign; and \\((-2)^2=4\\). Therefore the expression becomes \\(16+2(-8)-3(4)+5=16-16-12+5\\). Next, \\(16-16=0\\), \\(0-12=-12\\), and \\(-12+5=-7\\). Option A, 7, has the wrong final sign. Option B, -7, matches the calculation. Option C, 23, may result from mishandling the negative odd-power term or subtraction. Option D, 9, also does not follow the correct substitution and sign rules. Always use brackets around a negative substituted value. Memory cue: even powers become positive, odd powers retain the negative sign.

Correct answer: C. 3a^2b+7ab^2

Explanation: Only like terms can be added or subtracted. Here, 9a^2b and -6a^2b are like terms, so they give 3a^2b. Similarly, -4ab^2 and 11ab^2 give 7ab^2. Hence, the simplified expression is 3a^2b+7ab^2. Option A results from adding the coefficients of a^2b incorrectly. Exam tip: Before combining terms, check that both the variables and their exponents are identical.

Correct answer: C. 6

Explanation: displaystyle (2k+1)x-4x=[(2k+1)-4]x=(2k-3)x. Since its simplified form is 9x, the coefficients of x must be equal: 2k-3=9. Thus, 2k=12 and k=6. For example, choosing 5 gives coefficient 7, not 9. Exam tip: while combining like terms, add or subtract their coefficients only.