Algebraic expressions

Combine like terms: 2a^2-a^2=a^2, -3ab+5ab=2ab, and 4b^2-b^2=3b^2. Therefore, the simplified form is a^2+2ab+3b^2. In option B, the coefficients of the a^2 and b^2 terms have been added incorrectly. Exam tip: combine only terms with exactly the same variables and exponents.

Substituting z=-2, we get z^4=(-2)^4=16 and z^3=(-2)^3=-8. Therefore, z^4+z^3-z=16+(-8)-(-2)=16-8+2=10. Hence, 10 is correct. The nearby distractor 8 may result from incorrectly treating the final term -z as -2. Exam tip: an even power of a negative number is positive, while an odd power is negative.

Using the distributive property, \(5(2x-1)=10x-5\). Adding 3 gives \(10x-5+3=10x-2\). Therefore, the correct expression is \(10x-2\). \(10x+2\) results from incorrectly combining \(-5\) and 3. Exam tip: When opening brackets, multiply the outside number by every term inside the bracket.

Substitute -3 for x: \(x^2-2x+4=(-3)^2-2(-3)+4\). This gives \(9+6+4=19\), so 19 is correct. Option 7 may result from incorrectly treating \(-2(-3)\) as -6; the product of two negative numbers is positive. Exam tip: Always use brackets when squaring or substituting a negative value.

Combine like terms in the expression: \(6a-2a=4a\) and \(4b+5b=9b\). Therefore, the simplified form is \(4a+9b\). In \(8a+9b\), \(6a\) and \(-2a\) have been added incorrectly. Exam tip: Add or subtract coefficients only for terms with the same variable and the same power.

The square of the number \(p\) is \(p^2\), and four times the number is \(4p\). Since the question asks to add four times the number to its square, the expression is \(p^2+4p\). Option B uses subtraction, while option D multiplies the entire sum by 4. Exam tip: For phrases such as “add to,” use \(+\) and write the square and the multiple separately first.

The like terms containing \(m^2\) are \(9m^2\) and \(-3m^2\). Adding their coefficients gives \(9-3=6\), so the result is \(6m^2\). The term \(7m\) cannot be combined with them because its power of \(m\) is 1, not 2. Exam tip: combine terms only when both the variable and its exponent are the same.

Given \(a=2\) and \(b=-4\), \(ab+3a=(2)(-4)+3(2)=-8+6=-2\). Therefore, option A is correct. The value \(-14\) may result from incorrectly treating \(3a\) as \(-6\). Exam tip: after substitution, check the signs carefully while multiplying and adding negative numbers.

A minus sign before the bracket changes the sign of every term inside it: \(8x-(3x-5)=8x-3x+5\). Combining like terms gives \(8x-3x=5x\), so the simplified expression is \(5x+5\). The option \(5x-5\) results from incorrectly keeping the sign of \(-5\) unchanged. Exam tip: when removing brackets preceded by ‘−’, reverse every sign inside the brackets.

\(4x^2-3x+7\) has three terms: \(4x^2\), \(-3x\), and \(7\), so it is a trinomial. \(5a-2\) has only two terms. Exam tip: count terms separated by + or − signs.

Combine like terms: \(2y^2+3y^2=5y^2\) and \(5y-2y=3y\). The constant term \(-7\) remains unchanged. Therefore, the simplified expression is \(5y^2+3y-7\). In option C, the sign of the constant term is incorrect. Exam tip: add or subtract only terms with the same variable and the same exponent.

In the expression \(12p^2q\), no exponent is written on \(q\). Therefore, \(q=q^1\), so the power of \(q\) is \(1\). The exponent \(2\) belongs to \(p\), not to \(q\). Exam tip: A variable with no written exponent always has exponent \(1\).

The sum of \(x\) and \(y\) is \(x+y\). Taking 6 less than this sum means subtracting 6 from it, giving \(x+y-6\). The expression \(x+y+6\) adds 6, whereas the question asks for 6 less. Exam tip: For “less than” in this form, subtract the stated number from the expression already formed.

Expanding the brackets gives 4(3n+2)=12n+8 and -2(n+5)=-2n-10. Therefore, 12n+8-2n-10 = 10n-2. Option B results from using the wrong sign for the constant term in the second bracket. Exam tip: multiply the number before a negative bracket by both terms inside it.

Both (3ab^2) and (-10ab^2) have the variable part ab^2: the power of a is 1 and the power of b is 2. Only their coefficients, 3 and -10, are different, so they are like terms. Options B and D are incorrect because like terms may have different coefficients, but their variables and respective powers must match. Exam tip: Ignore coefficients first, then compare every variable and its exponent to identify like terms.

Substituting t=5 gives t^2-4t+6=5^2-4(5)+6=25-20+6=11. Therefore, 11 is the correct answer. The value 9 may result from incorrectly omitting the final +6. Exam tip: substitute the value first, then follow the order of powers, multiplication, and addition/subtraction.

Here, \(15xy\) and \(6xy\) are like terms, so their sum is \(21xy\). Similarly, \(-4x\) and \(9x\) are like terms and add to \(5x\). Therefore, the simplified expression is \(21xy+5x\). In option B, the sign while combining the \(x\)-terms is incorrect. Exam tip: Add or subtract only terms that have exactly the same variables with the same powers.

In the expression
\(-6x^2+11x-3\), the term containing
\(x^2\) is
\(-6x^2\). Therefore, the coefficient of
\(x^2\) is
\(-6\). Option 6 is incorrect because the negative sign is part of the coefficient. Exam tip: Include the sign when identifying the numerical factor multiplying a variable term.

Twice \(r\) is \(2r\), and three times \(s\) is \(3s\). Adding them gives \(2r+3s\), so option A is correct. In \(3r+2s\), the coefficients are interchanged, while expanding \(2(r+3s)\) gives \(2r+6s\). Exam tip: Write the stated multiple with each variable first, then use \(+\) for “sum.”

Substituting the given values, \(u^2-uv=(-2)^2-(-2)(1)=4-(-2)=4+2=6\). Hence, the correct answer is 6. The value 4 represents only \(u^2\); subtracting \(uv=-2\) gives 6. In exams, remember that subtracting a negative quantity changes it to addition.