Algebraic expressions

The sum of (x) and (y) is (x+y). Twice this entire sum is 2(x+y). In 2x+y, only x is multiplied by 2, not y. Exam tip: Use brackets when words such as “sum of” or “entire sum” occur.

In the expression \(7-3v\), \(v\) is the variable because its value can change. Here, \(7\) is the constant term and \(-3\) is the coefficient of \(v\), so \(-3\) is not a variable. Exam tip: Identify the letter or symbol whose value can change.

The governing concept is recognising algebraic operations from symbols. The plus sign in a + b indicates addition, so the expression is a sum. When a and b are written next to each other as ab, the implied operation is multiplication, so ab is the product of a and b. Therefore option A correctly states the difference. The expressions are not always equal; for example, if a = 2 and b = 3, then a + b = 5 whereas ab = 6. Option C reverses the meanings of the symbols, and option D is wrong because a and b may be variables rather than fixed constants.

The square of (x) is (x^2), and adding (3) gives (x^2+3). Understanding the order of words is important.

A constant term is a term with no variable. In 4x^2+3x+2, both 4x^2 and 3x contain x, whereas 2 contains no variable. Therefore, 2 is the constant term. Although x^0 equals 1, it is not a term in the given expression. Exam tip: identify the term that has no variable before selecting the constant term.

Substituting p=3 gives 4p-1=4×3-1=12-1=11. Therefore, the correct answer is 11. Option 12 is a close distractor if the subtraction of 1 is missed. Exam tip: After substitution, perform multiplication before addition or subtraction.

Here, 2x, 3x, and 4x are like terms because each term contains x to the power 1. So, add their coefficients: 2+3+4=9. Hence, the simplified expression is 9x. The option 24x comes from multiplying the coefficients, which is not required here. Exam tip: Only terms with the same variable and the same exponent can be added.

Zero is the additive identity: adding 0 to any number or variable does not change its value. Therefore, \(x+0=x\), so \(x\) is correct. \(1\) and \(x^0\) generally represent 1, so they are not the simplified form of \(x+0\). Exam tip: In addition, a term added with 0 can be removed.

1 is the multiplicative identity. Hence, multiplying any variable by 1 leaves it unchanged: \(1\cdot x=x\). The close distractor \(x+1\) is wrong because it adds 1 rather than multiplying by 1. Exam tip: multiplying an expression by 1 does not change it.

The product of 0 and any number or variable is 0. Therefore, \(0\cdot y=0\), and \(0\cdot y+6=0+6=6\). The option \(y+6\) is incorrect because the term containing \(y\) becomes 0 when multiplied by 0. Exam tip: simplify multiplication first, then perform addition or subtraction.

\(x+7\) is an algebraic expression because it contains the variable \(x\), the constant 7, and an operation. \(x=7\) is an equation, while \(x+7>0\) is an inequality; they contain relation symbols and are not merely expressions. Exam tip: An expression generally does not contain relation symbols such as \(=, <, >\).

In (5x), the number multiplying (x) is (5). To identify a coefficient, look at the number attached to the variable.

A constant term is a term that contains no variable. In 3a+4, the term 3a contains the variable a, whereas 4 has no variable. Therefore, 4 is the constant term. The number 3 is the coefficient of 3a, not the constant term. Exam tip: Identify the term with no letter or variable to find the constant term.

The expression \(2m+6\) has two terms separated by the plus sign: \(2m\) and \(6\). Therefore, it has 2 terms. In \(2m\), 2 is the coefficient and \(m\) is the variable; they are not separate terms. Exam tip: Count only the parts separated by + or − signs.

A variable is a letter or symbol whose value can change. In (6y-2), the value of y can vary, so y is the variable. Here, 6 is the coefficient of y and 2 is a constant; 8 does not occur in the expression. Exam tip: identify letters as variables and standalone numbers as constants.

Given x = 2, substitute 2 for x in the expression x + 5. This gives 2 + 5 = 7, so the correct answer is 7. Option 5 is only the constant term; the value of x must also be added. Exam tip: To evaluate an expression, first replace each variable with its given value and then simplify.

The expression \(x^2+3x-5\) has three terms: \(x^2\), \(3x\), and \(-5\), so it is a trinomial. \(4a-7\) is a binomial, \(9y\) is a monomial, and \(p+q+r+s\) has four terms. Exam tip: count the parts separated by + or − signs to find the number of terms.

In \(9-r\), the minus sign separates the terms. The two terms are \(9\) and \(-r\), so the correct answer is 2. It is not one term because its two parts are separated by a minus sign. Exam tip: Count each part separated by a \(+\) or \(-\) sign as a term.

4x and 3x are like terms because both contain x to the power 1. Therefore, add their coefficients: \(4x+3x=(4+3)x=7x\). The option 7 is only the sum of the coefficients and incorrectly leaves out x. Exam tip: Only terms with the same variable and the same exponent can be added or subtracted.

\(8b\) and \(5b\) are like terms because both contain \(b\) to the first power. Subtract their coefficients: \(8b-5b=(8-5)b=3b\). \(13b\) would result from addition, while \(3\) incorrectly omits the variable \(b\). Exam tip: add or subtract only the coefficients of like terms.