Algebraic expressions

By the zero multiplication property, multiplying any number or algebraic expression by 0 gives 0. Therefore, \(x\cdot0=0\). The value would be \(x\) if it were multiplied by 1, since \(x\cdot1=x\). Exam tip: If any factor in a product is 0, the entire product is 0.

By the multiplicative identity property, multiplying a variable by 1 does not change it: \(1\cdot a=a\). Therefore, \(1\cdot a+5=a+5\). \(6a\) is not correct because \(a\) and 5 are unlike terms and cannot be added. Exam tip: first simplify any multiplication by 1 in an expression.

A square has four equal sides. If each side is \(x\), its perimeter is \(x+x+x+x=4x\), so option B is correct. \(x^2\) represents the area of the square, not its perimeter. Exam tip: remember that the perimeter of a square is \(4 \times\) side.

The perimeter of a rectangle is the total length around its boundary. A rectangle has two equal lengths and two equal breadths, so its perimeter is found by adding l+3+l+3. This can also be written as twice the sum of one length and one breadth: P=2(l+3). The symbol l is an algebraic variable, so it should remain in the expression.

Substituting the given dimensions directly gives P=2(l+3)=2l+6. Option A gives only one length plus one breadth and is therefore half the perimeter. Option B is an area-like product and does not represent a boundary length, while option C has an unsuitable squared term. Hence option D, 2(l+3), is the correct expression.

The price of one pen is \(p\) rupees. Therefore, the total price of 5 pens is \(5\times p=5p\) rupees. \(p+5\) represents adding 5 rupees to the price, whereas here the price per pen must be multiplied by 5. Exam tip: Total cost of identical items = cost per item × number of items.

To find an age after 4 years, add 4 to the present age. Therefore, Riya’s age will be \(a+4\) years. The expression \(a-4\) represents her age 4 years earlier, so it is incorrect. Exam tip: use addition for “after” and subtraction for “before.”

First, the sum of \(x\) and \(3\) is \(x+3\). To take twice this entire sum, multiply the bracketed expression by \(2\): \(2(x+3)\). In \(2x+3\), only \(x\) has been doubled, so it is not correct. Exam tip: When you see “twice the sum,” put the complete sum inside brackets.

The difference of 10 and m means subtract m from 10. Therefore, the expression is \(10-m\). The expression \(m-10\) means subtracting 10 from m, so it reverses the order. Exam tip: “the difference of a and b” is generally written as \(a-b\).

The coefficient is the numerical factor multiplying the specified variable term. Here, in the term 6x^2, the factor multiplying x^2 is 6; therefore, the coefficient of x^2 is 6. The number 1 is the constant term, while x is a term with a different power. Exam tip: First identify the term with exactly the variable and exponent asked for.

A constant term is a term containing no variable. The expression x² − x has two visible terms: x² and −x. Both contain x, so neither is a constant term. When a polynomial has no separately written number-only term, its constant term is taken to be 0. Therefore option C is correct. Option A is the quadratic term, and option B is the variable term with coefficient −1; neither is constant. Option D would be correct only if a standalone 1 appeared in the expression, but no such term is present. The expression may be written as x² − x + 0 to make the missing constant explicit.

\(6a\), \(2a^2\), and \(-5\) are three separate terms because they are separated by + or − signs. Hence it is a trinomial. Exam tip: combine only like terms before counting terms.

In the expression ab + 2, a and b are two distinct variables, so option D is correct. The expressions 9x + 2 and m² + 3 each have only one variable, while 7 is a constant. Exam tip: Count the distinct letters in an expression to identify the number of variables.

In \(x^2\), the variable \(x\) has power 2 because 2 is written as its exponent. In \(2x\), 2 is the coefficient of \(x\), not its power; the power of \(x\) there is 1. In exams, look for the number written as a superscript on the variable.

An algebraic expression can contain numbers, variables, and operations. Its terms are the separate parts that are added or subtracted. For example, in an expression such as \(3x^2-5x+7\), the terms are \(3x^2\), \(-5x\), and \(+7\). The plus or minus sign belongs with the term that follows it, so the middle term is properly read as negative 5x.

Multiplication signs may occur inside a term, as in \(4xy\), but they do not usually separate terms. Division signs also do not separate terms in the usual identification of terms, and an equal sign compares two expressions rather than dividing one expression into terms. Therefore addition and subtraction are the main signs used to separate terms. Option C is correct, and the supplied explanation appropriately reminds us to keep the sign attached to its term.

Combining like terms gives \(3x-2x=x\), and combining the constants gives \(4+1=5\). Therefore, the simplified expression is \(x+5\). In \(x+3\), the constant terms have been added incorrectly. Exam tip: Add or subtract only like terms, such as \(x\)-terms with \(x\)-terms and constants with constants.

Substitute the given values: \(2a+b=2\times2+3=4+3=7\). Therefore, the correct answer is 7. Getting 6 may result from an error in the order of multiplication and addition. Exam tip: substitute the values of variables first, then perform multiplication before addition.

The term 5x^2y can be written as 5 \(\times x^2y\). Thus, the numerical factor multiplying \(x^2y\) is 5, so its coefficient is 5. The number 2 is the exponent of x in \(x^2\), not the coefficient. Exam tip: The number multiplying the stated algebraic part of a term is its coefficient.

Half of \(x\) is \(\frac{x}{2}\). “3 more than” means adding 3 to that quantity, so the correct expression is \(\frac{x}{2}+3\). \(\frac{x+3}{2}\) represents half of \(x+3\), so it is different. Exam tip: For “more than,” write the stated quantity first, then add the given number.

“2 more than c” means adding 2 to c, so the correct expression is \(c+2\). In contrast, \(c-2\) means 2 less than c, and \(2c\) means twice c. Exam tip: the phrase “more than” indicates addition.

Given \(x=0\), substitute it into \(7x+2\): \(7(0)+2=0+2=2\). Therefore, the correct value is \(2\). The number \(7\) is only the coefficient, not the value of the expression. Exam tip: To evaluate an expression, substitute the given value of the variable everywhere it occurs.

The governing concept is translating verbal mathematical language into algebraic notation while preserving grouping. The phrase “the sum of m and n” means m + n. The phrase “three times” means that the entire sum is multiplied by 3, so the required expression is 3(m + n). Therefore option C is correct. The brackets are essential because they show that 3 applies to both terms; using the distributive law gives 3(m + n) = 3m + 3n. Option A multiplies only m, and option B multiplies only n, so neither represents three times the complete sum. Option D contains the product mn and an added constant 3, which expresses a different operation altogether.

The governing concept is the numerical coefficient of a term. A coefficient is the number that multiplies the variable or literal part of an algebraic term. In 9x²y, the literal part is x²y and the numerical factor attached to it is 9. Thus the numerical coefficient is 9, making option B correct. Option A is the variable part, not a numerical value. Option C is the exponent of x; an exponent tells how many times x is used as a factor and is not the coefficient of the whole term. Option D, 9x, is only part of the term and still contains a variable, so it cannot be called the numerical coefficient. Separating coefficient, variable, and exponent prevents these common confusions.

Given \(x=2\), substitute it in \(4x+3\): \(4(2)+3=8+3=11\). Therefore, the correct answer is 11. The value 8 is only \(4\times2\); the additional 3 must still be added. Exam tip: Substitute the value of the variable first, then follow the order of operations.

\(6a\) and \(-2a\) are like terms, so their coefficients are subtracted: \(6a-2a=4a\). The constant term \(+5\) remains unchanged. Therefore, the simplified form is \(4a+5\). \(8a+5\) would result from adding \(6a\) and \(2a\), not subtracting them. Exam tip: Add or subtract only terms with the same variable and exponent.

The product of \(m\) and \(n\) is \(mn\). Subtracting 7 from this product gives \(mn-7\), so option D is correct. In \(7-mn\), the product \(mn\) is subtracted from 7, which is a different expression. Exam tip: In phrases such as “subtract from,” write the quantity from which subtraction is made first.