Algebraic expressions

Expand the brackets first: \(3(2x-5)=6x-15\) and \(-2(x+4)=-2x-8\). Combining like terms gives \(6x-15-2x-8=4x-23\). Therefore, the correct answer is \(4x-23\). The result \(4x-7\) can arise from an error while multiplying \(-2\) by \(4\). Exam tip: when a bracket is preceded by a negative coefficient, apply it carefully to every term inside the bracket.

Given \(a=-2\), we have \(a^2=(-2)^2=4\). Substituting in the expression, \(4a^2-3a-5=4(4)-3(-2)-5=16+6-5=17\). Therefore, 17 is correct. The value 9 can result from handling the sign of \(-3a\) incorrectly. Exam tip: Always use brackets around a negative substituted value before squaring or multiplying.

Direct answer: Option C, \(3p^2+5p+3\). To simplify, combine only like terms: terms with \(p^2\) go together, terms with \(p\) go together, and constants go together. First, \(5p^2-2p^2=3p^2\). Next, \(-3p+8p=5p\). Finally, \(7-4=3\). Putting these results together gives \(3p^2+5p+3\). Option A has the wrong coefficient of \(p^2\) and also the wrong constant. Option B changes the signs and does not correctly combine the terms. Option C is correct because every like-term calculation is correct. Option D incorrectly adds the two quadratic coefficients and gives the wrong coefficient of \(p\). A common mistake is combining unlike terms; \(p^2\), \(p\), and constants must remain separate.

Changing signs of the subtracted expression gives (2x+3-5x+9=-3x+12). Change signs carefully while removing brackets.

Substituting the given values, \(2x^2-xy+4y=2(3)^2-(3)(-1)+4(-1)\). Therefore, \(18+3-4=17\), so 17 is correct. Since \(y=-1\), the term \(-xy\) becomes \(-(3\times-1)=+3\); mishandling this sign can lead to the nearby option 21. Exam tip: always substitute a negative value using brackets.

If \(n\) is even, the three consecutive even numbers are \(n\), \(n+2\), and \(n+4\). Their sum is \(n+(n+2)+(n+4)=3n+6\). Therefore, option B is correct. \(n+6\) adds 6 only to the first number; it does not represent the sum of all three numbers. Exam tip: consecutive even numbers always differ by 2.

On opening the brackets, \(7m-2(3m-4)+5(m-1)=7m-6m+8+5m-5\). Combining like terms gives \((7m-6m+5m)+(8-5)=6m+3\). Therefore, the correct answer is \(6m+3\). The option \(6m-13\) results from handling the signs of the constant terms incorrectly. Exam tip: When a minus sign or coefficient occurs before brackets, multiply it by every term inside the brackets.

The direct answer is C: 20. Substitute the given value carefully. First, if r=4, then r squared is about multiplying 4 by 4, so r²=16. Next, calculate the two terms in the numerator: 3r²=3×16=48 and 2r=2×4=8. Subtract: 48−8=40. Finally divide the complete numerator by 2: 40÷2=20, or \\(\frac{3(4)^2-2(4)}{2}=\frac{48-8}{2}=20\\). A is wrong because 16 is only r², not the value of the full expression. B is wrong because it does not result from the stated substitution and simplification. D is wrong because the correct numerator gives 40, which becomes 20 after division. C is correct. Memory cue: substitute first, simplify the whole numerator, then divide.

\(4a^2b\) and \(3a^2b\) are like terms, so their sum is \(7a^2b\). Similarly, \(-7ab^2\) and \(2ab^2\) add to \(-5ab^2\). Therefore, the simplified expression is \(7a^2b-5ab^2\). In option C, the coefficients of \(a^2b\) have not been added correctly. Exam tip: combine only terms that have exactly the same variables with the same powers.

In \(x^2-7\), the \(x^2\) term is present, but there is no first-degree \(x\) term. Therefore, the coefficient of \(x\) is \(0\). In \(2x^2+x\), the coefficient of \(x\) is \(1\), so it is not correct. Exam tip: If a term of a particular degree is absent, its coefficient is \(0\).

\(2(x-y)-3(y-x)=2x-2y-3y+3x\), because \(-3(y-x)=-3y+3x\). Combining like terms gives \(5x-5y\). \(5y-5x\) is the negative of the required expression, so it is not correct. Exam tip: When a bracket is preceded by a minus sign, change the sign of every term inside it.

Given u=-3 and v=2, we have u²=9, uv=(-3)(2)=-6, and v²=4. Hence, u²-2uv+v²=9-2(-6)+4=9+12+4=25. The value 13 can result from handling the sign of the negative product uv incorrectly. Exam tip: determine the sign of uv first, then evaluate the complete term -2uv.

The perimeter of a rectangle is \(2(\text{length}+\text{breadth})\). Thus, \(2[(2x+3)+(x-5)] = 2(3x-2) = 6x-4\). Therefore, \(6x-4\) is correct. \(3x-2\) is only the sum of the length and breadth, not the perimeter. Exam tip: Always multiply the sum of length and breadth by 2 for a rectangle’s perimeter.

Using distribution, \((x+3)(x-3)=x^2-3x+3x-9=x^2-9\). Rima used the wrong sign for \(3\times(-3)\). Exam tip: apply \((a+b)(a-b)=a^2-b^2\) to avoid expansion errors.

On expanding, \(4(2a-3b)=8a-12b\) and \(-3(a-2b)=-3a+6b\). Therefore, \(8a-12b-3a+6b=5a-6b\). In option C, \(-3a\) has incorrectly been added instead of subtracted. Exam tip: When a negative multiplier is outside brackets, apply it to every term inside the brackets.

For x=1, we have x^3=1 and x^2=1. Therefore, 5x^3-4x^2+3x-2 = 5(1)-4(1)+3(1)-2 = 5-4+3-2 = 2. Hence, the correct answer is 2. Getting 4 may result from an error in addition or subtraction. Exam tip: After substitution, write each term with its sign carefully, especially the negative signs.

Direct answer: Option A, \((x+y)^2-xy\). The phrase “the square of the sum of x and y” must first be translated as \((x+y)^2\), because the whole sum is inside the brackets and then squared. The phrase “subtract xy from it” means take away \(xy\), so the complete expression is \((x+y)^2-xy\). Option A follows both instructions in the correct order. Option B is \(x^2+y-xy\), which does not square the complete sum and does not even square y. Option C is \(x+y^2-xy\), which squares only y, not the sum. Option D reverses the subtraction and gives \(xy-(x+y)^2\), the negative of the required expression. The brackets are essential: \((x+y)^2\) is different from \(x+y^2\). Memory cue: translate words step by step—sum, square, then subtract.

First simplify the innermost bracket: 2x-(x-5)=2x-x+5=x+5. Then, 3x-(x+5)=3x-x-5=2x-5. Hence, the correct answer is 2x-5. In 2x+5, the negative sign before the outer bracket has not been applied to every term inside it. Exam tip: When a bracket is preceded by a minus sign, change the sign of each term inside while removing the bracket.

Substituting t=-1 gives 2(-1)^3-3(-1)^2+4(-1)+6. Thus, -2-3-4+6=-3, so the correct answer is -3. The option 3 can result from incorrectly handling the negative signs in (-1)^3 or 4(-1). Exam tip: an odd power of a negative number is negative, while an even power is positive.

The governing concept is simplifying an algebraic expression by applying the distributive property and combining like terms. First simplify the innermost product: 4(1 − x) = 4 − 4x. Thus the square bracket becomes 3x + 4 − 4x = 4 − x. The original expression is now 9x − 2(4 − x). Distribute −2 across the parentheses: −2(4 − x) = −8 + 2x. Finally combine like terms: 9x + 2x − 8 = 11x − 8. Therefore, option A is correct. A common error is to lose the negative sign before 2, which can produce 7x − 8 or a positive constant. The brackets must be simplified from the inside outward.

Given a+b=7 and ab=10, substitute these directly into 2(a+b)-ab: 2(7)-10=14-10=4. Therefore, the correct option is 4. The value 14 is only the value of 2(a+b); ab must still be subtracted. In exams, treat a grouped expression such as (a+b) as one unit before substituting.

\(3x^4-2x^2+7\) has three terms, and the highest exponent of x is 4, so it is a trinomial polynomial of degree 4. Option B has four terms, while D has a negative exponent and is not a polynomial. Exam tip: check term count and highest exponent separately.

Substituting the given values, \,\((x+y)^2-xy=(2+3)^2-(2\times3)=5^2-6=25-6=19\). Therefore, the correct answer is 19. The value 25 is only \,\((x+y)^2\); \,\(xy=6\) must also be subtracted. Exam tip: Evaluate the squared term first, then perform multiplication and subtraction in order.

In
-6x^2+5x-1, the term
-6x^2 can be written as
(-6)x^2. Therefore, the coefficient of
x^2 is
-6. In option A, the coefficient is 6, while in option D the coefficient of
x^2 is 1;
-6 is the coefficient of x there. Exam tip: To identify a coefficient, include the sign and consider the complete numerical factor multiplying the term.

Write (\frac{x}{2}) as (\frac{2x}{4}), then the sum is (\frac{5x}{4}). Make common denominators for fraction terms.