Algebraic expressions

Combining like terms gives \(8x^2-5x^2=3x^2\) and \(-7xy+4xy=-3xy\). Therefore, the simplified expression is \(3x^2-3xy\). The terms \(x^2\) and \(xy\) are not like terms because their variable parts differ, so they cannot be combined. Exam tip: Add or subtract coefficients only when both the variables and their powers are exactly the same.

Given \(m-n=9\). Factoring 2 from the first two terms of \(2m-2n+11\) gives \(2(m-n)+11\). Therefore, \(2(9)+11=18+11=29\). Hence, 29 is correct. Option 31 would result from incorrectly adding \(18+11\). Exam tip: When a value of a group such as \(m-n\) is given, factor the expression to form that exact group before substituting.

On expanding, \(4(3x-2y)=12x-8y\) and \(-3(x-4y)=-3x+12y\). Therefore, the expression becomes \(12x-8y-3x+12y+5y=9x+9y\). In option C, the coefficients of \(x\) have been added incorrectly. Exam tip: When a bracket is multiplied by a negative number, apply the negative sign to every term inside it.

While adding polynomials, combine only like terms. Here, \(2x^2+3x^2=5x^2\), \(5x-7x=-2x\), and \(-6+10=4\). Therefore, the sum is \(5x^2-2x+4\), so option C is correct. Option A incorrectly adds the \(x\)-terms. Exam tip: add the \(x^2\)-terms, \(x\)-terms, and constant terms separately.

When x = 0, \(x^3\), \(x^2\), and \(x\) all become 0. Thus, \(-4(0)^3+9(0)^2-5(0)+13=13\). Therefore, 13 is correct. The number 9 is only the coefficient of \(x^2\), not the value of the expression. Exam tip: At x = 0, the value of a polynomial is its constant term.

First simplify inside the square bracket: \(3a-4(a-2)=3a-4a+8=-a+8\). Then \(5a-2[-a+8]=5a+2a-16=7a-16\). Therefore, the correct answer is \(7a-16\). The expression \(7a+16\) results from incorrectly handling the sign of \(+8\) when multiplying by \(-2\). Exam tip: When a negative coefficient is outside brackets, distribute it to every term carefully.

Substituting the given values: (s^3t-st^3+2t) = 2^3(-1)-2(-1)^3+2(-1) = -8+2-2 = -8. Therefore, the correct answer is -8. One may get -6 by making an error while adding the final term, 2t. Exam tip: An odd power of a negative number, such as (-1)^3, remains negative.

Apply the distributive law and simplify from the innermost grouping. First, 4x - {3x + 1} = 4x - 3x - 1 = x - 1. Also, 6(x - 2) = 6x - 12. Substituting these results gives (6x - 12) - (x - 1). The outer subtraction changes the signs of both terms in the second bracket: 6x - 12 - x + 1 = 5x - 11. Therefore, option A is correct. Option B incorrectly keeps the constant as -13, while options C and D use 7x, usually caused by adding rather than subtracting the x term in the second bracket.

First simplify inside the square bracket: \(4x-(3x+1)=4x-3x-1=x-1\). Then the full expression becomes \(6(x-2)-(x-1)=6x-12-x+1=5x-11\). Hence, \(5x-11\) is correct. \(5x-13\) results from handling the sign of \(-1\) incorrectly while subtracting \((x-1)\). Exam tip: When a minus sign precedes brackets, change the sign of every term inside them.

The number 7 less than \(x\) is \(x-7\), and the number 4 more than \(x\) is \(x+4\). Their sum is \((x-7)+(x+4)=2x-3\), so option A is correct. \(2x+11\) would result from adding 7 as well, but “7 less” requires subtraction. Exam tip: write “less than” as subtraction and “more than” as addition before simplifying.

(\frac{x}{5}=\frac{2x}{10}), so (\frac{7x}{10}-\frac{2x}{10}=\frac{x}{2}). Make a common denominator for fraction terms.

Given \(x+y=8\) and \(x-y=2\), substitute these values directly into the expression: \(4(x+y)-5(x-y)=4\times8-5\times2=32-10=22\). The value 24 may result from handling the subtraction term \(5(x-y)\) incorrectly. Exam tip: When values of grouped expressions such as \(x+y\) and \(x-y\) are given, substitute them directly as complete units.

Combining (x^2), (xy), and (y^2) terms separately gives (3x^2+5xy-5y^2). Do not mix different variable parts.

The governing concept is distribution of a negative factor and removal of brackets. First simplify the product: 3(2 - x) = 6 - 3x. Hence the square-bracket expression becomes 4x - (6 - 3x) = 4x - 6 + 3x = 7x - 6. The full expression is therefore 15 - (7x - 6). Applying the outer minus sign gives 15 - 7x + 6 = 21 - 7x. Thus option B is correct. Option A drops the coefficient created by combining 4x and 3x; options C and D arise from mishandling the outer subtraction or changing the sign pattern incorrectly.

Substituting \(t=3\), we get \(2t^4-5t^3+t^2=2(3^4)-5(3^3)+3^2\). Since \(3^4=81\), \(3^3=27\), and \(3^2=9\), the value is \(2\times81-5\times27+9=162-135+9=36\). Therefore, \(36\) is correct. A value such as \(18\) can result from incorrectly evaluating powers or coefficients. Exam tip: calculate powers first, then perform multiplication and subtraction.

The sum is (a+b), and five times it is (5(a+b)). Subtracting the difference (a-b) gives (5(a+b)-(a-b)).

Like terms must have identical variables with identical exponents. Here, \(5a^2b=a^2b^1\), but \(8ab=a^1b^1\); the power of \(a\) differs. Coefficients may differ in like terms. Exam tip: compare every exponent.

Substitute \(p=-1\): \(p^4=1\), \(p^3=-1\), and \(p^2=1\). Therefore, \(2p^4-3p^3+4p^2-5=2(1)-3(-1)+4(1)-5=2+3+4-5=4\). Hence, option B is correct. \(2\) is only the value of the first term, \(2p^4\), not of the whole expression. Exam tip: an even power of \(-1\) is \(1\), while an odd power is \(-1\).

In \((6x-3y)-(2x+8y)\), the sign of every term in the second expression changes: \(6x-3y-2x-8y\). Combining like terms gives \(6x-2x=4x\) and \(-3y-8y=-11y\), so the result is \(4x-11y\). In option C, the sign of \(8y\) has not been changed correctly. Exam tip: When a minus sign appears before brackets, change the signs of all terms inside the brackets.

Combine like terms: \(4x^2y+5x^2y=9x^2y\) and \(-9xy^2+6xy^2=-3xy^2\). Hence, the simplified form is \(9x^2y-3xy^2\). The terms \(x^2y\) and \(xy^2\) are not like terms because the powers of \(x\) and \(y\) differ, so they cannot be combined. Exam tip: before adding terms, check that every variable and its exponent match.