Algebraic expressions

When a whole bracket is subtracted, the minus sign must be applied to every term inside that bracket. Thus \\(-(x^2+5x-4)=-x^2-5x+4\\). The expression becomes \\(3x^2-2x+7-x^2-5x+4\\). Now combine like terms: \\(3x^2-x^2=2x^2\\), \\(-2x-5x=-7x\\), and \\(7+4=11\\).

Hence the simplified expression is \\(2x^2-7x+11\\), so option B is correct. The most common error is to change the sign of only the first term in the bracket. Since the subtraction applies to the entire bracket, every sign inside changes. Option A would result from mishandling the signs of the linear and constant terms.

Substitute \(x=-1\) into every occurrence of x, keeping the powers and signs clear. Since \((-1)^4=1\) and \((-1)^3=-1\), the expression becomes \(2(-1)^4-3(-1)^3+(-1)-5\). This is \(2(1)-3(-1)-1-5=2+3-1-5=-1\). Therefore, option D is correct. Parentheses are important when substituting a negative value, especially into an odd power.

The sign in the second term needs careful attention: the term is \(-3x^3\), so after substitution it is \(-3(-1)=+3\). The fourth-power term is positive because an even power of -1 is 1, whereas the third-power term is -1. Adding all terms gives \(-1\). The other listed values do not result from the correct substitution, so D follows directly from the calculation.

Only like terms can be added or subtracted. The coefficients of the \(a^2b\) terms give \(6-4=2\), and those of the \(ab^2\) terms give \(2+9=11\). Hence, the simplified expression is \(2a^2b+11ab^2\). In option A, the \(a^2b\) terms have incorrectly been added instead of subtracted. Exam tip: before combining terms, check that both the variables and their exponents are exactly the same.

Both terms contain the same variable \(x\), so add their coefficients: \((k-2)x+5x=(k-2+5)x=(k+3)x\). Since the simplified expression is \(9x\), we get \(k+3=9\). Therefore, \(k=6\). If 9 were chosen, the coefficient would become 12, not 9. Exam tip: while combining like terms, add only their coefficients.

Add the two given equations: \((x-y)+(x+y)=8+14\). The terms \(-y\) and \(+y\) cancel, giving \(2x=22\). Hence, \(x=11\). Option 22 is the value of \(2x\), not of \(x\). Exam tip: In a pair of linear equations, add equations when one variable has opposite signs so that it is eliminated.

Using the distributive property, \(4(3p-2q)=12p-8q\) and \(-2(5p+q)=-10p-2q\). Thus, the expression becomes \(12p-8q-10p-2q+7q\). Combining like terms gives \(12p-10p=2p\) and \(-8q-2q+7q=-3q\), so the result is \(2p-3q\). In \(2p+3q\), the sign of the \(q\)-term is incorrect. Exam tip: when a negative multiplier is outside brackets, apply it to every term inside the brackets.

Substituting m=2 and n=-3, we get 2m²n = 2×(2²)×(-3) = -24 and mn² = 2×(-3)² = 18. Therefore, 2m²n + mn² = -24 + 18 = -6. The nearby option 6 can result from incorrectly handling the sign of the negative term. Exam tip: the square of a negative number is positive, so (-3)² = 9.

“The square of the sum of (a) and (b)” is \((a+b)^2\). Subtracting \(3ab\) from it gives \((a+b)^2-3ab\), so option C is correct. In option D, the order of subtraction is reversed, so its value is generally different. Exam tip: In phrases such as “subtract X from Y,” write Y first and then subtract X.

In \(3x^4-2x+7\), the powers of x are 4, 1 and 0, all non-negative integers. Since \(1/x=x^{-1}\), option B is not a polynomial. Exam tip: reject expressions with negative or fractional powers.

First simplify the inner bracket: \(2x-(4x+3)=2x-4x-3=-2x-3\). Now, \(5x-(-2x-3)=5x+2x+3=7x+3\). Therefore, the correct answer is \(7x+3\). In \(7x-3\), the sign of the constant term has been changed incorrectly. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.

At x=2, the expression is equal to 9. So, k(2)^2-5(2)+3=9, or 4k-10+3=9. Hence, 4k-7=9, giving 4k=16 and k=4. If k=3, the expression evaluates to 5, not 9. Exam tip: In value-based questions, substitute the given variable value carefully, especially in terms with exponents.

Like terms have exactly the same variables and powers. In the first option, every term has variable part (x^2yz).

The governing concept is subtraction of algebraic expressions by distributing the negative sign across every term in the bracket. Start with 2a^2 + 3ab − b^2 − (a^2 − 4ab + 2b^2). The bracket becomes −a^2 + 4ab − 2b^2. Combining like terms gives (2a^2 − a^2) + (3ab + 4ab) + (−b^2 − 2b^2) = a^2 + 7ab − 3b^2. Hence option B is correct. The other options result from retaining or mishandling one or more signs, especially the negative before 4ab or 2b^2.

Given \(x+y=9\) and \(xy=14\), substitute these values directly: \(2(x+y)+3xy=2(9)+3(14)=18+42=60\). Therefore, option C is correct. \(42\) is only the value of \(3xy\); the term \(2(x+y)\) must also be added. Exam tip: When \(x+y\) and \(xy\) are given directly, substitute them into the expression without finding \(x\) and \(y\) separately.

Expand each bracket first: \(3r(r+2)=3r^2+6r\). Also, \(-2r(4-r)=-8r+2r^2\), since \(-2r\times(-r)=+2r^2\). Combining like terms gives \(3r^2+2r^2+6r-8r=5r^2-2r\). Hence, \(5r^2-2r\) is correct. The option \(5r^2+14r\) results from handling the negative sign incorrectly. Exam tip: carefully track signs while expanding brackets.

Combine like terms: \(7p^2q-4p^2q=3p^2q\) and \(-2pq+5pq=3pq\). Thus, the expression becomes \(3p^2q+3pq+9\), which has three terms: \(3p^2q\), \(3pq\), and \(9\). Note that \(p^2q\) and \(pq\) are not like terms because the exponent of \(p\) is different. Exam tip: combine only terms with identical variables and exponents.

In \(m^2n^3\), the exponent of \(m\) is 2 and the exponent of \(n\) is 3, so option B is correct. In \(m^3n^2\), the exponents are interchanged, making it a close but incorrect option. Exam tip: Read the exponent written on each variable separately.

Given \(x=-2\) and \(y=4\), \(x^2y=(-2)^2\times4=4\times4=16\). Also, \(xy^2=(-2)\times4^2=(-2)\times16=-32\). Therefore, \(x^2y-xy^2=16-(-32)=16+32=48\). The option \(-48\) may result from incorrectly handling the subtraction of a negative number. Exam tip: evaluate powers first, then multiply, and carefully track signs.

Add terms with the same power of \(x\): \(6x^2-2x^2=4x^2\), \(-3x+7x=4x\), and \(2-9=-7\). Hence, the sum is \(4x^2+4x-7\). In option D, the coefficients of \(x^2\) have been added incorrectly: \(6+(-2)=4\), not 8. Exam tip: While adding polynomials, group like terms first and carefully retain negative signs.

Given 3x+2=17, subtracting 2 from both sides gives 3x=15, so x=5. Hence, x^2-1=5^2-1=25-1=24. Option 25 is only the value of x^2; 1 still has to be subtracted. Exam tip: first find the value of the variable, then substitute it into the complete expression.