Algebraic expressions

Correct answer: D. \((k+3)^2\)

Explanation: The number 3 more than k is \(k+3\). Squaring this entire quantity gives \((k+3)^2\), so option D is correct. In \(k^2+3\), 3 is added only to the square of k; it is not the square of \(k+3\). Exam tip: For “square of” a quantity, write the complete quantity in brackets before applying the exponent 2.

Correct answer: A. \(3x^2-5x+7\)

Explanation: In \(3x^2-5x+7\), the powers of x are 2, 1 and 0, all non-negative integers. It has three terms, so it is a trinomial polynomial. \(4x-9\) is only a binomial. Exam tip: reject fractional or negative powers first.

Correct answer: B. 14

Explanation: Given \(x+y=6\). Taking 3 as a common factor, \(3x+3y-4=3(x+y)-4\). Substituting the given value gives \(3(6)-4=18-4=14\). Therefore, the correct answer is 14. Option 18 is a close distractor because it results from forgetting to subtract 4. Exam tip: First rewrite terms with a common coefficient as a multiple of the given sum.

Correct answer: A. \(9xy+3x\)

Explanation: Combine like terms: \(6xy+3xy=9xy\) and \(-2x+5x=3x\). Therefore, the simplified expression is \(9xy+3x\). The terms \(xy\) and \(x\) are unlike terms, so they cannot be combined with each other. Exam tip: Add or subtract coefficients only when the variables and their powers are exactly the same.

Correct answer: D. 3a-5b

Explanation: Simplify from the innermost bracket: \(3b-\{a-2b\}=3b-a+2b=5b-a\). Then \(2a-[5b-a]=2a-5b+a=3a-5b\). Hence, the correct answer is \(3a-5b\). The option \(3a-b\) results from missing the sign change of \(-2b\) when the inner bracket is subtracted. Exam tip: When a bracket is preceded by a minus sign, change the sign of every term inside it.

Correct answer: C. \(2x+11\)

Explanation: Expand the brackets: \(3(2x+1)-4(x-2)=6x+3-4x+8\). Combining like terms gives \(6x-4x=2x\) and \(3+8=11\). Therefore, the simplified form is \(2x+11\). \(2x-5\) results from incorrectly taking \(-4\times -2\) as \(-8\) instead of \(+8\). Exam tip: when a negative coefficient multiplies a bracket, check the sign of every term carefully.

Correct answer: A. \(2a-6b\)

Explanation: Subtraction means adding the opposite of every term in the second expression: \((5a-2b)-(3a+4b)=5a-2b-3a-4b\). Combining like terms gives \(5a-3a=2a\) and \(-2b-4b=-6b\). Therefore, the result is \(2a-6b\). In \(2a+2b\), the sign of \(4b\) has been handled incorrectly. Exam tip: when a minus sign precedes parentheses, change the signs of all terms inside them.

Correct answer: D. -11

Explanation: For x=-2, x^3=(-2)^3=-8 and x^2=(-2)^2=4. Therefore, x^3-2x^2+5=-8-2(4)+5=-8-8+5=-11. Hence, -11 is correct. The answer -3 may result from mishandling the sign or multiplication in the term -2x^2. Exam tip: the square of a negative number is positive, while its cube is negative.

Correct answer: B. \(9p+4q\)

Explanation: On expanding the brackets, \(2(3p-q)=6p-2q\) and \(3(p+2q)=3p+6q\). Combining like terms gives \(6p+3p=9p\) and \(-2q+6q=4q\). Therefore, the simplified form is \(9p+4q\). In \(9p-4q\), the signs of the \(q\)-terms have been combined incorrectly. Exam tip: multiply the number outside each bracket by every term inside it.

Correct answer: C. (9-(t^2+2t))

Explanation: Here, the entire sum (t^2+2t) is to be subtracted from 9. Therefore, the correct expression is (9-(t^2+2t)). In option A, only t^2 is subtracted; the sign of 2t should also change because the whole sum is being subtracted. Exam tip: In “subtract ... from ...”, write the first quantity minus the complete second expression.

Correct answer: A. 20

Explanation: Given \(m+n=8\) and \(m-n=2\), substitute these directly: \(3(m+n)-2(m-n)=3\times 8-2\times 2=24-4=20\). Therefore, 20 is correct. A value such as 22 can result from not subtracting \(2(m-n)\) correctly. Exam tip: Treat each given bracketed expression as one quantity, multiply first, and then subtract.

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

Explanation: Add like terms: \(2x^2+x^2=3x^2\), \(-3x+5x=2x\), and \(1+(-4)=-3\). Therefore, the sum is \(3x^2+2x-3\). In option A, the coefficients of the \(x\)-terms have been combined with an incorrect sign. In exams, align like terms by degree before adding their coefficients.

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

Explanation: First simplify inside the square bracket: \(2x+\{3-x\}=2x+3-x=x+3\). Then \(6x-[x+3]=6x-x-3=5x-3\). Therefore, the correct answer is \(5x-3\). Writing \(5x+3\) is incorrect because the minus sign before the square bracket changes \(+3\) to \(-3\). Exam tip: When removing a bracket preceded by a minus sign, change the signs of all terms inside it.

Correct answer: B. \(\frac{3}{x}+2\)

Explanation: In a polynomial, each variable has a zero or positive integer exponent. Since \(\frac{3}{x}=3x^{-1}\), the exponent of \(x\) is \(-1\), so it is not a polynomial. \(\sqrt{2}\) may be a coefficient. Exam tip: a variable in the denominator indicates a non-polynomial.

Correct answer: B. \(6x+5\)

Explanation: The perimeter of a triangle is the sum of its three sides: \((x+2)+(2x-1)+(3x+4)\). Combining like terms gives \(x+2x+3x=6x\) and \(2-1+4=5\). Therefore, the perimeter is \(6x+5\). In \(6x+7\), the constant terms have been added incorrectly. Exam tip: Add variable terms and constant terms separately.

Correct answer: A. \(-2y+19\)

Explanation: Using the distributive property, \(-3(2y-5)=-6y+15\) and \(4(y+1)=4y+4\). Therefore, \(-6y+15+4y+4=-2y+19\). Hence, \(-2y+19\) is the correct simplified form. Option \(2y+19\) results from an error while combining \(-6y\) and \(4y\). Exam tip: When multiplying a bracket by a negative number, check the sign of every term carefully.

Correct answer: A. \(9ab+2a\)

Explanation: Combine only like terms: \(7ab+2ab=9ab\) and \(-3a+5a=2a\). Therefore, the simplified form is \(9ab+2a\). The terms \(ab\) and \(a\) are not like terms because \(ab\) also contains the factor \(b\), so they cannot be combined. Exam tip: before adding terms, check that their variable parts and exponents are identical.

Correct answer: B. 2

Explanation: On substituting \(x=4\), \(x^2=4^2=16\) and \(2x=2\times4=8\). Therefore, \(\frac{x^2-2x}{4}=\frac{16-8}{4}=\frac{8}{4}=2\). Hence, option 2 is correct. Option 1 can result from an error while subtracting in the numerator or dividing by 4. Exam tip: after substitution, evaluate powers first, simplify the numerator, and then divide.

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

Explanation: Using the distributive property, \(2x(3x+1)=6x^2+2x\) and \(x(4x-5)=4x^2-5x\). Therefore, \((6x^2+2x)-(4x^2-5x)=6x^2+2x-4x^2+5x=2x^2+7x\). Option C results from handling the sign of the linear term incorrectly. Exam tip: when a bracket is preceded by a minus sign, change the signs of all terms inside it.

Correct answer: B. \(2(x-2)+5\)

Explanation: The difference of \(x\) and \(2\) is \(x-2\). Twice this entire difference is \(2(x-2)\), and adding 5 gives \(2(x-2)+5\). In option A, only \(x\) is doubled rather than the whole difference \(x-2\). Exam tip: Use brackets when a multiplier applies to an entire difference.

Correct answer: C. 23

Explanation: Given \(x-y=5\). First factor out 4 from the expression: \(4x-4y+3=4(x-y)+3\). Substituting gives \(4(5)+3=20+3=23\). Therefore, the correct answer is 23. The value 20 is only \(4(x-y)\); the constant term 3 must also be added. Exam tip: factor out the common coefficient before using a given algebraic relation.

Correct answer: A. \(-x\)

Explanation: Because of the minus sign, \(-(3x-2)=-3x+2\), and \(2(x-5)=2x-10\). Thus, \(8-3x+2+2x-10=-x\). Therefore, \(-x\) is the correct option. The option \(x\) may result from removing the negative sign incorrectly. Exam tip: When a bracket is preceded by a minus sign, change the sign of every term inside it.

Correct answer: A. a^2+2ab+3b^2

Explanation: Combine like terms: 2a^2-a^2=a^2, -3ab+5ab=2ab, and 4b^2-b^2=3b^2. Therefore, the simplified form is a^2+2ab+3b^2. In option B, the coefficients of the a^2 and b^2 terms have been added incorrectly. Exam tip: combine only terms with exactly the same variables and exponents.

Correct answer: C. 10

Explanation: Substituting z=-2, we get z^4=(-2)^4=16 and z^3=(-2)^3=-8. Therefore, z^4+z^3-z=16+(-8)-(-2)=16-8+2=10. Hence, 10 is correct. The nearby distractor 8 may result from incorrectly treating the final term -z as -2. Exam tip: an even power of a negative number is positive, while an odd power is negative.

Correct answer: A. \(7x-13\)

Explanation: The expression is \(4(3x-2)-5(x+1)\). Expanding the brackets gives \(12x-8-5x-5\). Combining like terms, \(12x-5x=7x\) and \(-8-5=-13\). Therefore, the simplified form is \(7x-13\). The option \(7x+13\) results from an incorrect sign while expanding \(-5(x+1)\), which must be \(-5x-5\), not \(-5x+5\). Exam tip: When a minus sign precedes a bracket, apply it to every term inside the bracket.