Algebraic expressions

In \(x^2+6\), there is no term containing \(x\) to the first power. Therefore, the coefficient of \(x\) is \(0\). The coefficient of \(x^2\) is \(1\), but that is not the coefficient of \(x\). Exam tip: While finding a coefficient, check the exact power of the variable.

The expression is \(8k-3(2k-1)\). Using the distributive property, \(-3(2k-1)=-6k+3\). Therefore, \(8k-6k+3=2k+3\). Hence, the correct answer is \(2k+3\). Choosing \(2k-3\) is a common error caused by missing that \(-3\times(-1)=+3\). Exam tip: when a negative coefficient is outside brackets, distribute it to every term carefully.

The perimeter of a rectangle is 2(length + breadth). So, 2[(x+4)+(x-1)] = 2(2x+3) = (4x+6). Therefore, option B is correct. (2x+3) is only the sum of the length and breadth, not the perimeter. Exam tip: Do not forget to multiply by 2 when finding a rectangle’s perimeter.

Using the distributive property, multiply \(x\) by each term inside the bracket: \(x(x+3)=x\cdot x+x\cdot3=x^2+3x\). Therefore, \(x^2+3x\) is correct. In \(x^2+3\), the term \(3\) has not been multiplied by \(x\). Exam tip: when expanding brackets, multiply the outside term by every term inside the bracket.

Substituting m=2 and n=4 gives mn+m^2=2×4+2^2=8+4=12. Therefore, the correct answer is 12. The value 8 represents only mn; m^2=4 must also be added. Exam tip: after substitution, evaluate powers first, then multiplication, and finally addition.

Variables written together show multiplication and the (+) sign shows addition. So (xy) and (x+y) are different expressions.

The perimeter of a rectangle is \(2(l+b)\). Thus, \(2[(2x+5)+(x-1)]=2(3x+4)=6x+8\). \(3x+4\) is only the sum of length and breadth, not the perimeter. Exam tip: write the formula first.

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.