Algebraic expressions

Like terms must have exactly the same variables with the same powers. Their numerical coefficients may be different, and their order may be written in the same or a different way. In \\(-4ab\\) and \\(9ab\\), both terms contain one \\(a\\) and one \\(b\\), so their variable part is exactly \\(ab\\). Only the coefficients, -4 and 9, are different.

Thus option C is the pair of like terms. The other pairs do not match in variable part: \\(xy\\) differs from \\(x\\), \\(m^2\\) differs from \\(m\\), and \\(p\\) differs from \\(q\\). Like terms can be added or subtracted directly, but unlike terms cannot be combined into one similar term merely because they contain letters or numbers.

Combine like terms: \(7x-2x=5x\), and the constant terms give \(4+9=13\). Therefore, the simplified expression is \(5x+13\). The option \(9x+13\) results from incorrectly ignoring the negative sign before \(2x\). Exam tip: Add or subtract only terms with the same variable and exponent.

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.