Algebraic expressions

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.

The governing concept is substitution in an algebraic expression. When a variable is assigned a value, replace that variable with the given number and then follow the indicated arithmetic operations. Here r = 5, so (r + 7) / 2 becomes (5 + 7) / 2. First add the numerator: 5 + 7 = 12. Then divide by the denominator: 12 / 2 = 6. Hence option C is correct. Option A could result from an incorrect addition or division, while option B simply repeats the given value of r and does not evaluate the expression. Option D may arise from adding 2 rather than dividing by 2. The fraction bar applies to the complete numerator r + 7.

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.

Expand the brackets first: \(3(2x-5)=6x-15\) and \(-2(x+4)=-2x-8\). Combining like terms gives \(6x-15-2x-8=4x-23\). Therefore, the correct answer is \(4x-23\). The result \(4x-7\) can arise from an error while multiplying \(-2\) by \(4\). Exam tip: when a bracket is preceded by a negative coefficient, apply it carefully to every term inside the bracket.

Given \(a=-2\), we have \(a^2=(-2)^2=4\). Substituting in the expression, \(4a^2-3a-5=4(4)-3(-2)-5=16+6-5=17\). Therefore, 17 is correct. The value 9 can result from handling the sign of \(-3a\) incorrectly. Exam tip: Always use brackets around a negative substituted value before squaring or multiplying.

Combining like terms gives (5p^2-2p^2=3p^2), (-3p+8p=5p) and (7-4=3). Group terms by powers.

Changing signs of the subtracted expression gives (2x+3-5x+9=-3x+12). Change signs carefully while removing brackets.

Substituting the given values, \(2x^2-xy+4y=2(3)^2-(3)(-1)+4(-1)\). Therefore, \(18+3-4=17\), so 17 is correct. Since \(y=-1\), the term \(-xy\) becomes \(-(3\times-1)=+3\); mishandling this sign can lead to the nearby option 21. Exam tip: always substitute a negative value using brackets.

If \(n\) is even, the three consecutive even numbers are \(n\), \(n+2\), and \(n+4\). Their sum is \(n+(n+2)+(n+4)=3n+6\). Therefore, option B is correct. \(n+6\) adds 6 only to the first number; it does not represent the sum of all three numbers. Exam tip: consecutive even numbers always differ by 2.

On opening the brackets, \(7m-2(3m-4)+5(m-1)=7m-6m+8+5m-5\). Combining like terms gives \((7m-6m+5m)+(8-5)=6m+3\). Therefore, the correct answer is \(6m+3\). The option \(6m-13\) results from handling the signs of the constant terms incorrectly. Exam tip: When a minus sign or coefficient occurs before brackets, multiply it by every term inside the brackets.

Putting (r=4) gives (\frac{3(16)-8}{2}=\frac{40}{2}=20). Simplify the whole numerator first in a fraction.