Algebraic expressions

In \(9-5x\), the term containing \(x\) is \(-5x\). The number multiplying \(x\) is \(-5\), so the coefficient of \(x\) is \(-5\). Choosing \(5\) would ignore the negative sign. Exam tip: always include the sign of the term while identifying a coefficient.

Terms in an expression are parts separated by addition or subtraction signs. Here, the terms are \(2p^2\), \(-3q\), and \(8\). Therefore, the expression has 3 terms. Counting only the variables \(p\) and \(q\) would be incorrect because the constant \(8\) is also a term. Exam tip: Count the parts separated by + and − signs to find the number of terms.

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.

The governing concept is addition of algebraic fractions with like variable parts. Fractions must first be expressed with a common denominator. The denominators 3 and 6 have least common denominator 6. Rewrite 2x/3 as 4x/6, while x/6 already has denominator 6. Then add the numerators: 4x/6 + x/6 = 5x/6. Hence option B is correct. Option A is not obtained by the required addition. Option C, 3x/9, simplifies to x/3, which is not generally equal to 5x/6. Option D introduces x² even though adding linear terms does not multiply the variables or change their exponent. The variable remains x, and only the numerical coefficients are combined after the denominators are equal.

When a variable is written without a visible exponent, its exponent is understood to be 1. In xy², x therefore has power 1 and y has power 2, exactly as required. Option B is correct. In x²y, the powers are 2 and 1; in x²y² both are 2; and in xy both powers are 1, so the other expressions do not satisfy both conditions.

The governing concept is translating mathematical language into an algebraic expression while preserving the order of operations. “The square of x” is x² and “the square of y” is y², so their sum is x² + y². The phrase “subtract 2xy from that sum” means begin with the complete sum and take away 2xy, giving x² + y² − 2xy. Therefore option C is correct. Option A uses a difference, x² − y², instead of a sum. Option B adds 2xy rather than subtracting it. Option D reverses the subtraction order and represents 2xy − (x² + y²), which is the negative of the required expression. The wording “from the sum” is the key to the correct order.

For a monomial containing more than one variable, the total degree is found by adding the exponents of all variables in that term. In option A, x²y³ has exponent 2 on x and exponent 3 on y, so its total degree is 2 + 3 = 5. In option B, x³y³ has degree 3 + 3 = 6. In option C, x⁴y² has degree 4 + 2 = 6. In option D, xy² can be written as x¹y², giving degree 1 + 2 = 3. Therefore only option A satisfies the condition. The question asks for the combined power, not the exponent of one variable alone, so simply noticing a large individual exponent would be incorrect.