Algebraic expressions

The two \(a^2\) terms are like terms, so \(a^2+a^2=2a^2\). The term \(a\) and the constant \(3\) are unlike terms, so they cannot be combined further. Hence, the simplified expression is \(2a^2+a+3\). In particular, \(a^2\) cannot be added to \(a\) to make \(2a\). Exam tip: combine only terms with the same variable and exponent.

Multiplying x by -3 gives -3x. Adding 8 afterwards gives the expression -3x+8. In option D, -3 multiplies both x and 8, producing -3x-24, so it is not correct. Exam tip: words such as “then adding” mean that the added number stays outside the multiplication.

Here, \(2x^2\) and \(5x^2\) are like terms, so their sum is \(7x^2\). Similarly, \(-3x\) and \(-x\) add to \(-4x\). Therefore, the simplified form is \(7x^2-4x\). In option C, the \(x^2\) terms have been added incorrectly. Exam tip: add or subtract only terms with the same variable and the same exponent.

Using the distributive property, multiply \(3a\) by each term inside the bracket: \(3a\times 2a=6a^2\) and \(3a\times(-1)=-3a\). Therefore, the expression becomes \(6a^2-3a\). Option A is incorrect because it does not use \(a\times a=a^2\). Exam tip: When multiplying terms, multiply the coefficients and also combine the powers of the variables correctly.

In 5mn + 2m, both terms, 5mn and 2m, contain variables. Therefore, there is no term consisting only of a number, so it has no constant term. In contrast, -7 is the constant term in 4y - 7. Exam tip: A term with no variable is called a constant term.

Substituting u=1 and v=-2, \(4u+v^2=4(1)+(-2)^2=4+4=8\). Therefore, the correct answer is 8. A common error is getting 6 by treating \((-2)^2\) as -4; the square of a negative number is positive. Exam tip: Always use brackets when squaring a negative number.

First, the difference of \(x\) and \(y\) is \(x-y\). Twice this entire difference is \(2(x-y)\), which equals \(2x-2y\). In \(2x-y\), only \(x\) is doubled, so it is not correct. Exam tip: For “twice the difference,” put the difference in brackets and multiply the whole bracket by 2.

Only like terms can be added or subtracted. Here, 10q^2 and q^2 are like terms, so 10q^2+q^2=11q^2. Similarly, -4q+6q=2q. Therefore, the simplified expression is 11q^2+2q. In 9q^2+2q, the coefficients of the q^2 terms have been combined incorrectly. Exam tip: Group terms according to the power of the variable before simplifying.

Multiplying any variable or number by 0 gives 0. Hence, \(0x=0\), so \(0\) is the correct option. Although \(x^0=1\) for \(x\ne 0\), it is not equal to \(0x\). Exam tip: a term with coefficient 0 contributes nothing to the value of an expression.

Reena is incorrect. Like terms must have the same variables with the same powers. In \(p^2q\) and \(pq^2\), the powers of \(p\) and \(q\) differ. Exam tip: compare exponents, not just the variables.

In \(2x^2y^3\), the factor containing \(y\) is \(y^3\), so the power of \(y\) is 3. The power of \(x\) is 2, and their sum, 5, gives the total degree of the term, not the power of \(y\). Exam tip: To find the power of a variable, look at the exponent written directly on that variable.

Half of x is \(\frac{x}{2}\). Adding 7 to this quantity gives \(\frac{x}{2}+7\), so option C is correct. In \(\frac{x+7}{2}\), both x and 7 are divided by 2, while \(7-\frac{x}{2}\) represents a difference rather than a sum. Exam tip: translate “half of x” first, then add 7 for “sum.”

Given \(x=5\), substitute it into the expression: \(\frac{x+3}{2}=\frac{5+3}{2}=\frac{8}{2}=4\). Therefore, the correct answer is 4. The value 8 is only the numerator \(x+3\); it must still be divided by 2. Exam tip: After substitution, simplify step by step and pay attention to the order of operations.

Combine like terms: \(3x-x=2x\) and \(2y+5y=7y\). Therefore, the simplified expression is \(2x+7y\). \(4x+7y\) is incorrect because the coefficient in \(3x-x\) is \(3-1=2\), not 4. Exam tip: Add or subtract only terms with the same variable and the same power.

Using the distributive property, \(2(x+4)=2\cdot x+2\cdot4=2x+8\). Therefore, \(2x+8\) and \(2\cdot x+2\cdot4\) are equal to the given expression. Also, by the commutative property of addition, \(8+2x=2x+8\). However, in \(2x+4\), the constant 4 has not been multiplied by 2, so it is not equal to the given expression. Exam tip: When expanding brackets, multiply the outside factor by every term inside the bracket.

For like terms, variables and their powers must be exactly the same. Here swapped powers make the terms unlike.

Substituting n=-1 gives 4n^2+3n-2=4(-1)^2+3(-1)-2. Since (-1)^2=1, the value is 4(1)-3-2=4-3-2=-1. Getting 1 usually results from using the wrong sign for 3(-1). Exam tip: Always use brackets when squaring a negative number.

Like terms must have exactly the same variables raised to the same powers; only their numerical coefficients may differ. In 2ab, 3ab, and 5ab, both a and b have power 1, so they are like ab. However, -4a has no b, so it is not a like term of ab. Exam tip: Compare the variable part and exponents, not the coefficients.

Using the distributive property, \(3(2x-3)=3\times2x-3\times3=6x-9\). Therefore, option A is correct. In option D, the constant term becomes \(+9\), whereas the required expression has \(-9\). Exam tip: When expanding brackets, multiply the outside number by every term inside the bracket.

Combining like terms gives \(x^2-x^2=0\) and \(-2x+5x=3x\). The constant term \(4\) remains unchanged, so the simplified expression is \(3x+4\). Option \(7x+4\) is incorrect because \(-2x+5x=3x\), not \(7x\). Exam tip: add or subtract only terms with the same variable and exponent.

The term containing x² is 4x². The number multiplying x² is 4, so the coefficient of x² is 4. Here, -3 is the coefficient of x, while 7 is the constant term. Exam tip: To identify a coefficient, look at the number and sign multiplying the required variable part.

A constant term is a term that contains no variable. In this expression, 6a contains a and -5b contains b, whereas 9 has no variable. Therefore, 9 is the constant term. Exam tip: identify the term with no letter or variable to find the constant term.

In an expression, terms are separated by plus (+) or minus (−) signs. Here the terms are 3p^2, 2p, and −8, so there are 3 terms. The coefficient 2 in 2p is not counted as a separate term. Exam tip: Count each complete part separated by + or − as one term.

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.