Algebraic expressions

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.

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.

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.

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.

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.

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.

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.

The governing concept is addition of algebraic fractions: fractions must first be rewritten with a common denominator, while like algebraic terms retain the variable x. The denominators 3 and 6 have least common denominator 6. Thus 2x/3 = 4x/6, and adding x/6 gives 4x/6 + x/6 = 5x/6. Therefore option B is correct. Option A would result from an incorrect subtraction or coefficient operation. Option C, 3x/9, simplifies to x/3 and is not equivalent in general. Option D introduces x² even though only linear terms are being added, so it violates the required algebraic operation.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.