Algebraic expressions

Using the distributive property, multiply 2 by each term inside the bracket: \(2\times 3y=6y\) and \(2\times(-5)=-10\). Therefore, \(2(3y-5)=6y-10\). In option A, only \(3y\) has been multiplied; \(-5\) must also be multiplied by 2. Exam tip: When expanding brackets, multiply the outside number by every term inside.

Substitute \(a=3\): \(2a^2-a+1=2(3^2)-3+1\). Evaluate the exponent first: \(3^2=9\). Thus, \(2\times9-3+1=18-3+1=16\). Therefore, 16 is the correct option. 18 is a close distractor because it may result from ignoring the final \(-3+1\). Exam tip: In algebraic expressions, evaluate powers first, then multiplication, and finally addition or subtraction.

Substituting the given values, (3x+y)=3(2)+5=6+5=11. Therefore, the correct answer is 11. Option 9 may result from incorrectly evaluating 3×2 as 4. Exam tip: after substitution, perform multiplication before addition.

First, the square of \(n\) is \(n^2\). Three times this is \(3n^2\), and adding 4 gives \(3n^2+4\). In \(3(n+4)^2\), \(n\) and 4 are added before squaring, so it represents a different expression. Exam tip: Write “square of” as an exponent first, then apply the stated multiplication and addition.

Here, \(5r^2\) and \(-2r^2\) are like terms, so their sum is \((5-2)r^2=3r^2\). The term \(6r\) has exponent 1 on \(r\), so it cannot be combined with the \(r^2\) terms. Therefore, the simplified form is \(3r^2+6r\). Writing \(9r^2\) is incorrect because \(6r\) cannot be added to an \(r^2\) term. Exam tip: combine or subtract only terms having the same variable and the same exponent.

A binomial is an algebraic expression with two unlike terms. In \(4x^2-7\), the terms are \(4x^2\) and \(-7\), so it is a binomial. \(x^2+x+1\) has three terms and is a trinomial, while \(9x\) has only one term and is a monomial. Exam tip: Count the terms separated by plus or minus signs.

\(-3xy\) and \(8xy\) are like terms because both have the same variable part, \(xy\). Adding their coefficients gives \(-3+8=5\), so \(-3xy+8xy=5xy\). The constant term \(-2\) remains unchanged. Therefore, the simplified expression is \(5xy-2\). Option \(5x-y-2\) is incorrect because the term \(xy\) cannot be split into separate \(x\) and \(y\) terms. Exam tip: Add or subtract only terms with identical variables raised to identical powers.

Substituting p=-2, we get p^2+3p=(-2)^2+3(-2)=4-6=-2. Therefore, the correct answer is -2. A common mistake is to take (-2)^2 as -4, but the square of a negative number is positive. Exam tip: evaluate powers first, then perform multiplication and addition or subtraction.

Combine like terms: \(9m+2m=11m\) and \(-4n-6n=-10n\). Therefore, the simplified expression is \(11m-10n\). In option B, the coefficients of \(m\) have been added incorrectly. Exam tip: Add or subtract only terms with the same variable and the same exponent.

First, the sum of \(2x\) and \(3\) is \(2x+3\). “5 less than the sum” means subtracting 5 from this sum, so the correct expression is \(2x+3-5\). \(5-(2x+3)\) would mean subtracting the sum from 5, which represents a different statement. Exam tip: In “less than” phrases, identify the quantity from which subtraction is to be made.

Using the distributive property, \(3(x+2)=3x+6\). Combining the like terms \(3x\) and \(4x\) gives \(7x\). Therefore, the simplified expression is \(7x+6\). The option \(3x+6\) incorrectly leaves out the outside \(4x\). Exam tip: Expand brackets first, then combine like terms.

The difference of \(a\) and \(b\) is \(a-b\). Four times this complete difference is \(4(a-b)\), so option C is correct. In \(4a-b\), only \(a\) is multiplied by 4; \(b\) must also be part of the multiplied difference. Exam tip: When a question says “times the difference,” put the entire difference in brackets before multiplying.

Substituting the given values, \(u-2v=1-2(-3)\). Since \(2(-3)=-6\), we get \(1-(-6)=1+6=7\). Therefore, the correct answer is 7. Option 5 may result from handling the subtraction of a negative number incorrectly. Exam tip: subtracting a negative quantity changes into addition.

Combine like terms: \(12c^2+3c^2=15c^2\) and \(-5c-2c=-7c\). Therefore, the simplified expression is \(15c^2-7c\). In \(15c^2-3c\), the linear terms \(-5c\) and \(-2c\) have been combined incorrectly. Exam tip: Add or subtract only terms with the same variable and exponent.

After removing the numerical coefficient (4), the variable part (x^2y) remains. View a term as coefficient and variable part.

\(2x^2\) and \(-5x^2\) are like terms because both have the variable part \(x^2\). Add their coefficients: \(2+(-5)=-3\). Therefore, the sum is \(-3x^2\). \(-3x^4\) is incorrect because exponents are not added when like terms are added. Exam tip: before adding or subtracting, check that the variable parts and their exponents are the same.

On expanding the brackets, \(5(a-2)=5a-10\) and \(-2(a+1)=-2a-2\). Therefore, \(5a-10-2a-2=3a-12\). Hence, the correct answer is \(3a-12\). \(3a-8\) results from handling the signs of the constant terms incorrectly. Exam tip: When a negative coefficient is outside a bracket, multiply it by every term inside the bracket.

Substituting x=-1, x^2-4x+3=(-1)^2-4(-1)+3=1+4+3=8. Therefore, the correct answer is 8. A common error is to treat -4(-1) as -4, but the product of two negative numbers is positive. Exam tip: Always use brackets when substituting a negative value.

In \(x^2+6\), there is no term containing \(x\) to the first power. Therefore, the coefficient of \(x\) is \(0\). The coefficient of \(x^2\) is \(1\), but that is not the coefficient of \(x\). Exam tip: While finding a coefficient, check the exact power of the variable.

The expression is \(8k-3(2k-1)\). Using the distributive property, \(-3(2k-1)=-6k+3\). Therefore, \(8k-6k+3=2k+3\). Hence, the correct answer is \(2k+3\). Choosing \(2k-3\) is a common error caused by missing that \(-3\times(-1)=+3\). Exam tip: when a negative coefficient is outside brackets, distribute it to every term carefully.

The perimeter of a rectangle is 2(length + breadth). So, 2[(x+4)+(x-1)] = 2(2x+3) = (4x+6). Therefore, option B is correct. (2x+3) is only the sum of the length and breadth, not the perimeter. Exam tip: Do not forget to multiply by 2 when finding a rectangle’s perimeter.

Using the distributive property, multiply \(x\) by each term inside the bracket: \(x(x+3)=x\cdot x+x\cdot3=x^2+3x\). Therefore, \(x^2+3x\) is correct. In \(x^2+3\), the term \(3\) has not been multiplied by \(x\). Exam tip: when expanding brackets, multiply the outside term by every term inside the bracket.

Substituting m=2 and n=4 gives mn+m^2=2×4+2^2=8+4=12. Therefore, the correct answer is 12. The value 8 represents only mn; m^2=4 must also be added. Exam tip: after substitution, evaluate powers first, then multiplication, and finally addition.

Variables written together show multiplication and the (+) sign shows addition. So (xy) and (x+y) are different expressions.

The perimeter of a rectangle is \(2(l+b)\). Thus, \(2[(2x+5)+(x-1)]=2(3x+4)=6x+8\). \(3x+4\) is only the sum of length and breadth, not the perimeter. Exam tip: write the formula first.