Algebraic expressions

The terms are 3x^2, 2x, and 5. The powers of x are 2 and 1 respectively, while the constant term 5 has power 0. Hence, the highest power is 2. Note that 3 is the coefficient of x^2, not its power. Exam tip: To find the highest power of a polynomial, look at the exponent of the variable, not the coefficient.

Option D, 11, is a constant because its value is fixed and it contains no variable. The expressions m, 2m, and m+2 contain the variable m, so their values can change when m changes. Exam tip: A term with no letter or variable is a constant.

A constant term is a term that contains no variable. In \((10x+3y-4)\), \(10x\) and \(3y\) contain variables, whereas \(-4\) has no variable. Therefore, the constant term is \(-4\), not just \(4\), because the negative sign is part of the term. Exam tip: always retain the sign while identifying a constant term.

6x and 2x are like terms, so their coefficients are added: 6x + 2x = 8x. The constant 1 has no like x-term, so it remains separate. Therefore, the simplified expression is 8x + 1. Option 8x incorrectly omits the constant term 1. Exam tip: Add or subtract only terms with the same variable and exponent.

\(9a\) and \(-3a\) are like terms, so their coefficients combine as \(9-3=6\). Hence, \(9a+2b-3a=6a+2b\). The term \(2b\) cannot be combined with the terms containing \(a\), because they have different variables. Exam tip: Add or subtract only terms with the same variables raised to the same powers.

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 and 3b + 2a are equivalent for every value of a and b, so option A is correct. Option B, 5ab, represents a product involving a and b and is not generally equal to the original sum. Option C also represents multiplication and changes the structure of the expression. Option D changes addition into subtraction, so it gives a different value in general. Equivalence depends on preserving both the operations and the terms, not merely on using the same letters.

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.