Algebraic expressions

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.

Add the like terms in the two expressions: 2x^2+x^2=3x^2, -5x+4x=-x, and 3+(-8)=-5. Therefore, the sum is (3x^2-x-5). Option B incorrectly adds the x-terms. Exam tip: while adding polynomials, combine only terms with the same variable and exponent.

While subtracting the second polynomial, change the sign of each of its terms: \((9x^2-4x+1)-(3x^2+2x-5)=9x^2-4x+1-3x^2-2x+5\). Combining like terms gives \(6x^2-6x+6\). Option A has an incorrect linear term because \(-4x-2x=-6x\). Exam tip: change every sign inside the polynomial being subtracted before removing the brackets.

Two algebraic terms are like terms only when the exponent of every corresponding variable is the same. In \(x^2y\), x has exponent 2 and y has exponent 1. In \(xy^2\), x has exponent 1 and y has exponent 2. Although both terms contain the same two variables, their powers do not match. Therefore they are unlike terms, so option B is correct.

The order of writing the variables is not the real issue; the exponents attached to each variable must be compared. For instance, \(4x^2y\) and \(-7x^2y\) would be like terms because both have the pattern \(x^2y\). But \(x^2y\) and \(xy^2\) have different variable parts and cannot be combined by ordinary addition. They are not constants, and multiplying their variable parts is not what the question asks. Hence B is the correct statement.

Given \(a=1\) and \(b=-2\), we have \(a^2=1\), \(b^2=4\), and \(b^3=-8\). Thus, \(4a^2b-3ab^2+b^3=4(1)(-2)-3(1)(4)+(-8)=-8-12-8=-28\). Therefore, \(-28\) is correct. The option \(-12\) is only the value of the second term, not of the complete expression. Exam tip: an odd power of a negative number is negative, while an even power is positive.

On expanding the brackets, \(2(4x-3y)=8x-6y\) and \(-5(x-y)=-5x+5y\). Thus, the expression becomes \(8x-6y-5x+5y+3y\). Combining like terms gives \((8x-5x)+(-6y+5y+3y)=3x+2y\). The distractor \(3x-8y\) results from incorrectly handling the sign of the \(y\)-term while expanding \(-5(x-y)\). Exam tip: When a negative coefficient multiplies a bracket, apply it to every term inside the bracket.

In \(x^2+kx+16\), the term containing \(x\) is \(kx\). The coefficient of \(x\) in this term is \(k\). Since the coefficient of \(x\) is given as \(12\), \(k=12\). Therefore, \(k-5=12-5=7\). Option \(12\) is the value of \(k\), not of \(k-5\). Exam tip: identify the factor multiplying the variable to find its coefficient.

The square of \(x\) is \(x^2\), so three times its square is \(3x^2\). Twice \(y\) is \(2y\). The phrase “the difference between A and B” means \(A-B\), so the required expression is \(3x^2-2y\). Option C reverses the order of subtraction and represents \(2y-3x^2\). Exam tip: Write the square first, then apply the numerical coefficient.

Combine like terms: \(4x^2-x^2=3x^2\), \(-3xy+5xy=2xy\), and \(2y^2-7y^2=-5y^2\). Hence, the simplified expression is \(3x^2+2xy-5y^2\). Option B uses an incorrect coefficient for \(x^2\). Exam tip: add or subtract only terms having the same variables with the same powers.

Substitute the value directly into the expression:

\(2x(x-5)+4x^2=2\times3\times(3-5)+4\times3^2\)

\(=6\times(-2)+4\times9=-12+36=24\). Therefore, the correct answer is 24. A value such as 30 may result from an error while calculating the negative term \(-12\). In exams, evaluate powers such as \(x^2\) first and check signs inside brackets carefully.

Only like terms can be added or subtracted. The \(x^2y\) terms are \(3x^2y\) and \(-5x^2y\), so their sum is \((3-5)x^2y=-2x^2y\). Therefore, the coefficient of \(x^2y\) is \(-2\). The terms \(2xy^2\) and \(8xy^2\) are not like terms of \(x^2y\), so they do not affect this coefficient. Exam tip: Before combining terms, check that both the variables and their powers are identical.

Using the distributive law, \(x(4x-9)=x\cdot4x-x\cdot9=4x^2-9x\). Therefore, option A is correct. Option B expands to \(4x^2-36x\), so it is not equivalent even though it looks similar. Exam tip: To check a factored expression, multiply each term outside the bracket by every term inside it.