Algebraic expressions

Using the distributive property, multiply 2 by each term inside the bracket: \(2\times x=2x\) and \(2\times 3=6\). Hence, \(2(x+3)=2x+6\), so option B is correct. In \(2x+3\), the 3 has not been multiplied by 2. Exam tip: When expanding brackets, multiply the outside number by every term inside.

Using the distributive property, multiply 3 by each term inside the bracket: \(3(y-2)=3\times y+3\times(-2)=3y-6\). Therefore, \(3y-6\) is correct. In \(3y-2\), 3 has been multiplied only by \(y\), which is incorrect. Exam tip: multiply the factor outside the bracket by every term inside it.

“4 more than \(x\)” means add 4 to \(x\), so the correct expression is \(x+4\). \(x-4\) represents 4 less than \(x\), while \(4x\) means 4 times \(x\). Exam tip: use addition for “more than” and subtraction for “less than.”

To find twice a number or variable, multiply it by 2. Hence, twice of n is 2 × n = 2n. The expression n + 2 means adding 2 to n, not doubling it. Exam tip: the word “twice” indicates multiplication by 2.

“5 less than \(t\)” means subtracting 5 from \(t\), giving \(t-5\). Therefore, option B is correct. In \(5-t\), the order of subtraction is reversed, while \(t+5\) means 5 more than \(t\). Exam tip: In phrases such as “less than,” subtract the stated number from the quantity that follows “than.”

The word ‘sum’ means addition. Adding 6 to x gives \(x+6\). In contrast, \(x-6\) represents the difference between x and 6, not their sum. Exam tip: Match words such as sum, difference, and product with +, −, and × respectively.

A product is obtained by multiplying quantities. Thus, the product of m and 3 is 3 × m = 3m. The expression m + 3 represents addition, not multiplication. Exam tip: In algebra, the multiplication sign is usually omitted between a number and a variable, so 3 × m is written as 3m.

Given \(x=1\) and \(y=2\), substitute these values in \(x+y\): \(x+y=1+2=3\). Therefore, 3 is the correct answer. Option 2 is only the value of \(y\), not the sum. Exam tip: When evaluating an expression, substitute the given value of every variable before simplifying.

Given x=6, x^2=6^2=6×6=36. Therefore, 36 is the correct answer. The value 12 is twice 6, but squaring means multiplying the number by itself. Exam tip: An exponent of 2 always means the number multiplied by itself.

A numerical coefficient is the number that multiplies the variable part of a term. In the term \\(2p^2\\), the variable part is \\(p^2\\), and the number placed before it is 2. The coefficient tells us how many times the variable expression is being taken. It is different from the variable part itself, so \\(p^2\\) is not the numerical coefficient.

Therefore, the numerical coefficient of \\(2p^2\\) is 2, which is option C. No calculation beyond identifying the multiplying number is needed. The exponent 2 belongs to the variable \\(p\\); it does not change the coefficient. Option A gives the variable part, while options B and D give numbers or variables that do not multiply the term as written.

The governing concept is the commutative property of addition: changing the order of addends does not change their sum. Thus 2a + 3b equals 3b + 2a for every value of a and b. Option B, 5ab, multiplies a and b and is not generally equal to the original sum. Option C also represents multiplication and changes both the operation and the structure. Option D changes the plus sign to a minus sign, so it gives a different expression except in special cases. Therefore option A is the only equivalent expression.

The expression 5(x) represents the multiplication of 5 and x. Therefore, its simplified algebraic form is 5x. The expressions x+5 and 5+x represent addition, not multiplication. Exam tip: When a number is written immediately before a variable or bracket, it usually indicates multiplication.

The expression \(ab+2\) has two terms: \(ab\) and \(2\). Since \(ab\) contains the variables \(a\) and \(b\), it is the variable term. The number \(2\) is a constant term because it has no variable. Although \(2ab\) contains variables, it is not a term of the given expression. Exam tip: Separate terms using the addition (+) or subtraction (−) signs.

To find half of a variable, divide it by 2. Therefore, half of \(r\) is \(\frac{r}{2}\), so option D is correct. \(2r\) means twice \(r\), while \(r+2\) and \(r-2\) add or subtract 2. Exam tip: translate “half” as division by 2.

A coefficient is the numerical factor that multiplies a variable or a product of variables. In the term \(-8xy\), the letters \(x\) and \(y\) form the variable part, written as \(xy\). The number attached to this variable part is \(-8\), including its negative sign. Thus the term can be viewed as \((-8)\times(xy)\).

Therefore option C is correct. The coefficient is not 8 because omitting the minus sign changes the term, and it is not \(xy\), which is the variable part itself. The number 1 would be the coefficient only if the term were simply \(xy\). Always include the sign when identifying a coefficient: in \(-8xy\), the coefficient of \(xy\) is \(-8\).

In \(3x^2-5x+7\), the powers of \(x\) are \(2\), \(1\), and \(0\). All are non-negative integers, so it is a polynomial. In option B, \(\frac{2}{x}=2x^{-1}\) has a negative exponent; option C contains \(x^{1/2}\), and option D contains \(x^{-3}\), so they are not polynomials. Exam tip: in a polynomial, every variable exponent must be \(0,1,2,\ldots\).

Three times the number r is 3r. Subtracting 2 from this result gives 3r-2. The option r-6 means subtracting 6 from r, so it does not represent the statement. Exam tip: In “subtract ... from ...”, write the quantity being subtracted after a minus sign.

Parts of an expression separated by addition or subtraction signs are called terms. Here the terms are \(2a^2\), \(3a\), \(-5\), and \(7b\). Therefore, there are 4 terms. The constant \(-5\) is also counted as a separate term. Exam tip: While counting terms, include the sign attached to each term.

5x and 2x are like terms because both contain x to the first power. Therefore, 5x + 2x = 7x, while the constant term -4 remains unchanged. Hence, the equivalent expression is 7x - 4. In 3x - 4, the coefficients have been combined incorrectly. Exam tip: Add or subtract only like terms.

A constant term is a term that contains no variable. In 6m^2 - 9, the term 6m^2 contains the variable m, while -9 has no variable. Therefore, -9 is the constant term. The number 6 is a coefficient, not the constant term. Exam tip: choose the term with no letter or variable to identify the constant term.