Algebraic expressions

If the first integer is \(n\), the three consecutive integers are \(n\), \(n+1\), and \(n+2\). Their sum is \(n+(n+1)+(n+2)=3n+3\). In \(3n+1\), the constant term is incorrect. Exam tip: Write consecutive numbers explicitly before adding them.

Combine like terms: \(2x^2-3x^2=-x^2\) and \(5x+x=6x\). Therefore, the simplified expression is \(-x^2+6x\). Terms containing \(x^2\) and \(x\) are unlike terms, so they cannot be combined with each other. Exam tip: Group terms according to their powers before simplifying.

A negative term is being subtracted, so subtraction changes to addition: \(4a^2b-(-7a^2b)=4a^2b+7a^2b=11a^2b\). These are like terms because the powers of \(a\) and \(b\) are the same. Option A uses the sign rule incorrectly, while option D incorrectly multiplies the exponents. Exam tip: When a minus sign precedes a negative term, change it to plus before simplifying.

Given \(x=3\), \(2(x+4)-x^2=2(3+4)-3^2=2\times7-9=14-9=5\). Therefore, the correct answer is 5. The value 7 can result from incorrectly evaluating \(x^2\) or missing the multiplication by 2. Exam tip: after substitution, evaluate powers first, then multiplication, and finally addition or subtraction.

Three times \(x\) is \(3x\), and twice \(y\) is \(2y\). Adding these terms gives \(3x+2y\), so option A is correct. In \(2x+3y\), the coefficients of \(x\) and \(y\) are reversed. Exam tip: translate “times” into multiplication and “sum” into addition.

On opening the bracket, \(10-3(2-z)=10-6+3z\), because multiplying \(-3\) by \(2\) gives \(-6\), while multiplying it by \(-z\) gives \(+3z\). Therefore, the simplified form is \(4+3z\). In \(4-3z\), the sign from multiplying \(-3\) and \(-z\) has been taken incorrectly. Exam tip: When a negative coefficient precedes a bracket, multiply it by every term inside the bracket.

In 3x^2y, the power of x is 2 and the power of y is 1, whereas in -2xy^2, the power of x is 1 and the power of y is 2. Like terms must have exactly the same variables with the same powers; only their coefficients may differ. Hence, these are unlike terms. A constant term has no variable. Exam tip: compare exponents carefully, since x^2y and xy^2 are not like terms.

The first number is \(x\). The number that is 6 more than it is \(x+6\). Therefore, their sum is \(x+(x+6)=2x+6\). The option \(x+6\) represents only the second number, not the sum of both numbers. Exam tip: For “more than,” add the stated amount first, then calculate what is asked.

Combine like terms: \(2x-5x=-3x\) and \(3y+y=4y\). Therefore, the simplified expression is \(-3x+4y\). \(7x+4y\) results from incorrectly treating \(-5x\) as positive. Exam tip: add or subtract only terms with the same variable and exponent.

The governing concept is the translation of words into algebraic operations while preserving their order. “Subtract 3 from x” means form x − 3, because the quantity from which subtraction is made comes first. The phrase “then multiply the result by 5” means that the entire quantity x − 3 must be multiplied by 5, giving 5(x − 3). Therefore, option C is correct. Option A, 5x − 3, multiplies x first and subtracts 3 afterward, so it changes the order. Option B is x − 15, which subtracts 15 rather than multiplying the difference by 5. Option D represents 3(x − 5), reversing the numbers and their roles.

Substituting the given values, \(a^2-b^2=4^2-2^2=16-4=12\). Therefore, the correct answer is 12. Option 8 can result from an incorrect calculation of the squares. Exam tip: calculate each square first, and then subtract the results.

Write \(x\) as \(\frac{2x}{2}\) so that both terms have the same denominator. Hence, \(\frac{x}{2}+x=\frac{x}{2}+\frac{2x}{2}=\frac{3x}{2}\). \(2x\) is a close but incorrect choice because fractional terms must first be written with a common denominator before adding. Exam tip: make denominators common before adding fractional algebraic terms.

Use the distributive property: \(4(2x-1)=4\times2x-4\times1=8x-4\). Adding 3 gives \(8x-4+3=8x-1\). Therefore, the correct answer is \(8x-1\). The result \(8x-7\) would come from incorrectly subtracting 3. Exam tip: when expanding brackets, multiply the outside number by every term inside the bracket.

In \((3x+2)+(x-5)\), combine like terms: \(3x+x=4x\) and \(2-5=-3\). Therefore, the sum is \(4x-3\). The option \(4x+7\) results from adding the constants with an incorrect sign. Exam tip: while adding polynomials, combine only terms with the same variable and exponent.

Subtraction gives \((7x-4)-(2x+3)\). The minus sign before the second bracket changes the sign of each term: \(7x-4-2x-3\). Combining like terms, \(7x-2x=5x\) and \(-4-3=-7\), so the result is \(5x-7\). The option \(5x-1\) results from incorrectly retaining the sign of \(+3\). Exam tip: when subtracting an expression in brackets, change the sign of every term inside the bracket.

Substituting x=2 gives x^3+x=2^3+2=8+2=10. Therefore, the correct answer is 10. The value 8 is only x^3; the additional +x, or +2, must also be included. Exam tip: after substitution, evaluate powers before addition or subtraction.

Use the distributive property: \((-2x)(3x-4)=(-2x)(3x)+(-2x)(-4)=-6x^2+8x\). Hence, option A is correct. In option C, the sign of the second term is incorrect because the product of two negative quantities is positive. Exam tip: While removing brackets, multiply the outside term by every term inside and check the signs carefully.

First, the sum of \(x\) and \(4\) is \((x+4)\). To square the entire sum, the exponent 2 must be placed outside the brackets, so the correct expression is \((x+4)^2\). In \(x^2+4^2\), the terms are squared separately; it is not the square of their sum. Exam tip: For “square of a sum,” put the complete sum in brackets before writing the exponent \(2\).

The term containing y is 3y. Since y is multiplied by 3, its coefficient is 3. The number 4 is the coefficient of z in the term 4z, not of y. In exams, first identify the term containing the specified variable.

On expanding the bracket, \(4(2x-3)=8x-12\). Adding \(x\) gives \(8x-12+x=9x-12\). Therefore, the correct expression is \(9x-12\). In \(9x+12\), the sign of the constant term is incorrect because \(4\times(-3)=-12\). Exam tip: while expanding brackets, check the sign of each product carefully.

On substitution, a² + 2b = (-2)² + 2(3) = 4 + 6 = 10. Therefore, the correct answer is 10. The option 8 may result from incorrectly adding only 2 instead of calculating 2b. Exam tip: Always use brackets when squaring a negative number; (-2)² equals 4, not -4.

The terms \(7xy\) and \(5xy\) are like terms, so their sum is \(12xy\). Similarly, \(-2x+3x=x\). Therefore, the simplified expression is \(12xy+x\). \(12x+y\) is incorrect because an \(xy\) term cannot be split into separate \(x\) and \(y\) terms. Exam tip: Add or subtract only terms with identical variable parts and powers.

The square of the number \(n\) is \(n^2\), and 5 times the number is \(5n\). “Subtracting 5 times the number from its square” means subtracting \(5n\) from \(n^2\), giving \(n^2-5n\). In \(5n-n^2\), the order of subtraction is reversed. Exam tip: In “subtract A from B,” write \(B-A\).

Combine the like terms containing p²: 3p² - 2p² = p² = 1p². Therefore, the coefficient of p² is 1. The terms -4p and p are linear terms, so they do not affect the coefficient of p². Exam tip: Always include the sign while identifying a coefficient.

Using the distributive property, \(5(2x-1)=10x-5\). Adding 3 gives \(10x-5+3=10x-2\). Therefore, the correct expression is \(10x-2\). \(10x+2\) results from incorrectly combining \(-5\) and 3. Exam tip: When opening brackets, multiply the outside number by every term inside the bracket.