Algebraic expressions

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).

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.

Like terms have exactly the same variables raised to the same powers; only their coefficients may differ. The terms 4a^3b^2 and -5a^3b^2 have the same variable part, a^3b^2, so option B is correct. In option A, the powers of a and b are interchanged, so it is not a like term. Exam tip: Ignore the coefficient and compare the variables and their exponents.

A minus sign before the second bracket changes the sign of every term inside it: \(4x^2-5xy+2y^2-x^2-3xy+6y^2\). Combining like terms gives \((4-1)x^2+(-5-3)xy+(2+6)y^2=3x^2-8xy+8y^2\). Option D results from incorrectly keeping the sign of the \(xy\) term positive. Exam tip: When a bracket is preceded by ‘−’, change every sign inside it before combining like terms.

Subtract the first equation from the second: \((a+b)-(a-b)=13-5\). This gives \(2b=8\), so \(b=4\). Option 5 is the given value of \(a-b\), not the value of \(b\). Exam tip: Add or subtract a pair of equations to eliminate one variable quickly.

The term containing x^2 is -5x^2. The number multiplying x^2 is its coefficient, so the coefficient is -5. The number 2 is the coefficient of the x^3 term, 2x^3, not of x^2. Exam tip: first locate the term with the required power, then identify the number multiplying it.

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In \(x^2y^4\), the exponent of \(x\) is 2 and that of \(y\) is 4. Hence, its total degree is \(2+4=6\), so option A is correct. The total degree of \(x^3y^2\) is \(3+2=5\), so it is not correct. Exam tip: For a monomial, add the exponents of all its variables to find the total degree.

In \(4x^3-2x+7\), the powers of \(x\) are \(3\), \(1\), and \(0\). All are non-negative integers, so it is a polynomial in \(x\). In option A, \(\frac{3}{x}=3x^{-1}\); option B has \(\sqrt{x}=x^{1/2}\); and option D has \(x^{-2}\). These contain negative or fractional powers, so they are not polynomials. Exam tip: in a polynomial, variable exponents can only be \(0,1,2,\ldots\).

Only like terms can be added or subtracted. \(5u^2v\) and \(2u^2v\) are like terms, so their sum is \(7u^2v\). Similarly, \(-3uv+6uv=3uv\). Therefore, the simplified expression is \(7u^2v+3uv-4\). In option B, \(5+2\) has incorrectly been taken as \(3\). Exam tip: before combining terms, check that both the variables and their exponents are identical.

On substituting \(x=-3\), the numerator is \((-3)^3+2(-3)^2-6=-27+18-6=-15\). Hence, \(\frac{-15}{-3}=5\). Therefore, option A is correct. An answer such as \(7\) can result from an error with the power of a negative number or the subtraction sign. Exam tip: always use brackets when substituting a negative value, especially in powers.