Algebraic expressions

A minus sign before the second bracket changes the sign of every term inside it: \(2x^2-3x+5-x^2-4x+7\). Combining like terms gives \((2x^2-x^2)+(-3x-4x)+(5+7)=x^2-7x+12\). Therefore, option A is correct. In option D, the constant term has been given the wrong sign. Exam tip: when a bracket is preceded by a minus sign, change the sign of every term inside it.

Substitute \(x=-2\): \(3x^3-2x^2+5x-1=3(-2)^3-2(-2)^2+5(-2)-1\). Since \((-2)^3=-8\) and \((-2)^2=4\), the value is \(-24-8-10-1=-43\). A value such as \(-35\) can result from mishandling powers or negative signs. Exam tip: check the signs of odd and even powers of a negative number separately.

Only like terms can be added or subtracted. The \(a^2b\) terms give \(4a^2b+5a^2b=9a^2b\), and the \(ab^2\) terms give \(-3ab^2+7ab^2=4ab^2\). Therefore, the simplified expression is \(9a^2b+4ab^2\). \(16a^3b^3\) is incorrect because it would result from multiplication, whereas this question requires addition of terms. Exam tip: Check both the variables and their exponents before combining terms.

In option A, opening the brackets gives \(3(2p-5)-(p+3)=6p-15-p-3=5p-18\). Hence, it is correct. Option B simplifies to \(7p-18\), since \(2(3p-9)+p=6p-18+p\). Exam tip: When removing brackets preceded by a minus sign, change the sign of every term inside the bracket.

Substituting the given values, \(m^2n=3^2\times(-2)=9\times(-2)=-18\), and \(2mn^2=2\times3\times(-2)^2=2\times3\times4=24\). Therefore, \(m^2n-2mn^2=-18-24=-42\). The option \(-18\) is only the value of the first term; it does not subtract the second term. Exam tip: the square of a negative number is positive, so \((-2)^2=4\).

On expanding the brackets, the expression becomes 2x+6y-6x+3y+4x. Combining like terms, the coefficient of x is 2-6+4=0, while the coefficient of y is 6+3=9. Hence, the simplified expression is 9y. The option 6x+9y results from combining the x-terms incorrectly. Exam tip: When a bracket is multiplied by a negative number, apply the sign change to every term inside it.

The terms \(kx\) and \(6x\) are like terms, so \(kx+6x=(k+6)x\). For this to equal \(11x\), we need \(k+6=11\). Therefore, \(k=5\). If \(k=6\), the expression becomes \(12x\), not \(11x\). Exam tip: add the coefficients when combining like terms.

Combining like terms gives \(7u^2v+3u^2v=10u^2v\) and \(-4uv^2-9uv^2=-13uv^2\). Thus, the simplified expression is \(10u^2v-13uv^2\), so the coefficient of \(uv^2\) is \(-13\). The value \(-4\) is the coefficient of only one term, not the combined like terms. Exam tip: first collect all terms with exactly the same variable part before identifying a coefficient.

Substituting the given values: \(2x^2-xy+y^2=2(1)^2-(1)(-3)+(-3)^2\). Thus, \(2+3+9=14\). Therefore, the correct answer is 14. Option 8 may result from incorrectly treating \(-xy\) as negative despite both \(x\) and \(y\) being multiplied with a negative sign. Exam tip: the square of a negative number is positive, and \(- (1\times -3)=+3\).

A minus sign precedes the second bracket, so the signs of all its terms change: \(5x^2-(2x^2-3x)+(4x-7)=5x^2-2x^2+3x+4x-7=3x^2+7x-7\). Therefore, the coefficient of \(x\) is \(7\). The number \(-7\) is the constant term, not the coefficient of \(x\). Exam tip: When a bracket is preceded by a minus sign, change the sign of every term inside it.

The direct answer is option A. Like terms must have exactly the same variables raised to exactly the same powers. Numerical coefficients may differ, but the variable part must match. In option A, \(3a^2b\), \(-5a^2b\), and \(8a^2b\) all have the variable part \(a^2b\), so they form a complete group of like terms. Option B contains \(a^2b\), \(ab^2\), and \(a^2\); the powers or variables differ, so they are not all alike. Option C contains \(2ab\), \(4a^2b\), and \(6ab^2\); again, the exponents do not match. Option D contains \(a^2b\), \(a^3b\), and \(a^2b^2\), which also have different powers. Option A is correct even though one coefficient is negative, because the sign and coefficient may change while the literal part remains identical. Exam cue: compare letters and exponents, not coefficients.

On expanding the brackets, \(3(2a-b)-2(a+3b)+5b=6a-3b-2a-6b+5b\). Combining like terms gives \(6a-2a=4a\) and \(-3b-6b+5b=-4b\). Therefore, the simplified expression is \(4a-4b\). The option \(4a+4b\) results from handling the negative signs of the \(b\)-terms incorrectly. Exam tip: when a minus sign precedes a bracket, change the sign of every term inside it.

The value of the expression is 0 when \(x=3\). So, \(2(3)^2+3k-15=0\). This gives \(18+3k-15=0\), hence \(3+3k=0\) and \(k=-1\). If \(k=1\), the expression equals 6, not 0. Exam tip: When a polynomial is stated to be zero at a given value of \(x\), substitute that value directly and solve the resulting equation.

Combining like terms gives \(4x^2y+2x^2y=6x^2y\) and \(-6xy+9xy=3xy\). Thus, the simplified expression is \(6x^2y+3xy-5\). It contains three different types of terms: an \(x^2y\) term, an \(xy\) term, and the constant term \(-5\). The terms \(6x^2y\) and \(3xy\) are not like terms because their variable parts are different. Exam tip: check both variables and their exponents before combining terms.

In (7x^2-3x), both terms contain variables and there is no constant term. It has two terms with different powers.

Use the distributive property: \(2p(p-3)=2p^2-6p\) and \(4(p+1)=4p+4\). Combining like terms gives \(2p^2-6p+4p+4=2p^2-2p+4\). Therefore, option A is correct. Option C incorrectly omits the \(4p\) term from the second bracket. Exam tip: after expanding brackets, combine only like terms with the same power of the variable.

Given \,\(x+y=5\), substitute 5 for the complete group \,\(x+y\) in \,\(2(x+y)-3\): \,\(2\times 5-3=10-3=7\). Therefore, the correct answer is 7. Option 10 results from finding \,\(2(x+y)\) but forgetting to subtract 3. Exam tip: When a grouped expression is given a value, substitute the value for the entire group.

The like terms containing \(r^2s\) are \(9r^2s\) and \(-5r^2s\). Thus, \(9r^2s-5r^2s=4r^2s\), so the coefficient of \(r^2s\) is 4. Although \(-4rs+rs=-3rs\), this is not a like term of \(r^2s\). Exam tip: Combine only terms with exactly the same variables raised to the same powers.

The phrase “adding \((x+y)\) to four times \((3x-2y)\)” means first multiply the complete expression \((3x-2y)\) by 4, and then add \((x+y)\). Hence, \(4(3x-2y)+(x+y)\) is correct. In option B, 4 multiplies the entire sum, so it represents a different expression. Exam tip: When a whole expression is multiplied, enclose it in brackets.

Add the like terms in the two polynomials: \(x^2+2x^2=3x^2\), \(2x-x=x\), and \(-3+4=1\). Therefore, the sum is \(3x^2+x+1\). The option \(3x^4+x+1\) is incorrect because powers are not added while adding polynomials; only coefficients of like terms are combined. Exam tip: Arrange terms in descending powers before combining like terms.

To subtract the second polynomial, change the sign of every term in it: \((5x^2-3x+8)-(2x^2+4x-1)=5x^2-3x+8-2x^2-4x+1\). Combining like terms gives \(3x^2-7x+9\). Hence, option B is correct. Option C results from an incorrect sign while subtracting \(4x\). Exam tip: a minus sign before brackets changes the sign of every term inside them.

Given a-b=4 and b=3, substitute b=3 to get a-3=4; hence a=7. Therefore, 2a-b=2(7)-3=14-3=11. Option 14 results from forgetting to subtract b. Exam tip: First find the value of the unknown variable by substitution, then evaluate the required expression.

Combining like terms gives \(3xy+4xy=7xy\) and \(-2x+7x=5x\). Thus, the simplified expression is \(7xy+5x-5y\), so \(9x\) does not remain. \(5x\) is a close distractor because it is correctly obtained by adding the two \(x\)-terms. Exam tip: combine only terms that have the same variables with the same powers.

First apply the distributive property: \(2x(x-4)=2x^2-8x\). Adding \(3x\) gives \(2x^2-8x+3x=2x^2-5x\). Therefore, option A is correct. In option B, \(3x\) has incorrectly been treated as the constant term \(3\). Exam tip: combine only like terms; \(-8x+3x=-5x\).

At \(x=2\), the expression is given to have value \(16\). So, \(2^2+2k+6=16\), which gives \(10+2k=16\). Hence, \(2k=6\) and \(k=3\). If \(k=2\), the value would be 14, not 16. Exam tip: Substitute the given value of \(x\) into every term before simplifying.