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Expanding Brackets Quiz 1 (30 MCQs)

This multiple-choice question set evaluates students' understanding of expanding brackets in algebraic expressions and equations, including combining like terms, applying the distributive property, and solving linear equations. It also covers factorization techniques such as finding the highest common factor (HCF) or greatest common divisor (GCD).

Quiz Instructions

Select an option to see the correct answer instantly.

1. Expand and simplify(y + 3) (y + 5)

Explanations: To expand and simplify $(y + 3)(y + 5)$, use the distributive property (also known as FOIL: First, Outer, Inner, Last): - First terms: $y \cdot y = y^2$ - Outer terms: $y \cdot 5 = 5y$ - Inner terms: $3 \cdot y = 3y$ - Last terms: $3 \cdot 5 = 15$ Combine these results: \[y^2 + 5y + 3y + 15 = y^2 + 8y + 15\] This matches option C.

Option Analysis:

Option A: Incorrect. No cubic term and incorrect constant.
Option B: Incorrect. Missing $y$ terms, incorrect constant.
Option C: Correct. Matches the simplified expression.
Option D: Duplicate of Option C, correct but listed twice.

2. Expand the following brackets:4(a + 2)

Explanations: To expand the brackets $4(a + 2)$, you need to distribute the number outside the parentheses, which is 4, to each term inside the parentheses. This means multiplying 4 by $a$ and then by 2. - First, multiply 4 by $a$: $4 \times a = 4a$. - Next, multiply 4 by 2: $4 \times 2 = 8$. Combining these results gives you the expression $4a + 8$, which is why Option A is correct.

Option Analysis:

Option A: Correct. $4(a + 2) = 4a + 8$.
Option B: Incorrect. Adding 6 instead of 8 after distributing the 4.
Option C: Incorrect. Only multiplying by $a$ and not by 2.
Option D: Incorrect. Adding 4 instead of 8 after distributing the 4.

3. Expand and Simplify(xy+3)(x+y)

Explanations: To expand and simplify $(xy + 3)(x + y)$, apply the distributive property (also known as FOIL for binomials): - First: $xy \cdot x = x^2y$ - Outer: $xy \cdot y = xy^2$ - Inner: $3 \cdot x = 3x$ - Last: $3 \cdot y = 3y$ Combining these, we get: \[x^2y + xy^2 + 3x + 3y\] This matches Option B.

Option Analysis:

Option A: Incorrect. It misses the $xy^2$ term and incorrectly duplicates the $xy$ term.
Option B: Correct. Matches the expanded form exactly.
Option C: Incorrect. Only includes a part of the expansion, specifically $3x$.
Option D: Incorrect. It incorrectly duplicates the $xy$ term and misses the $xy^2$ term.

4. Solve. x/4 + 11 = 20

Explanations: The claimed correct answer is D) 36 because the equation $ \frac{x}{4} + 11 = 20 $ can be solved by first subtracting 11 from both sides, resulting in $ \frac{x}{4} = 9 $. Multiplying both sides by 4 gives $ x = 36 $.

Option Analysis:

Option A: Incorrect. Solving the equation does not yield a fraction.
Option B: Incorrect. The solution is an integer, not a single digit.
Option C: Incorrect. This value would result in $ \frac{31}{4} + 11 = 20 $, which does not equal 20 when simplified.
Option D: Correct. Multiplying both sides of the equation by 4 after isolating $ \frac{x}{4} $ gives $ x = 36 $.

5. Expand and simplify the following quadratic expression ..... (x-6)(x-4)

Explanations: To expand and simplify the quadratic expression $(x-6)(x-4)$, we use the distributive property (also known as FOIL: First, Outer, Inner, Last). First: $x \cdot x = x^2$ Outer: $x \cdot -4 = -4x$ Inner: $-6 \cdot x = -6x$ Last: $-6 \cdot -4 = 24$ Adding these together: $x^2 - 4x - 6x + 24 = x^2 - 10x + 24$.

Option Analysis:

Option A: Incorrect. The correct expression does not have a positive coefficient for the $x$ term.
Option B: Incorrect. The constant term is incorrect; it should be positive 24, not negative 24.
Option C: Incorrect. The coefficient of the $x$ term is incorrect; it should be -10, not +10.
Option D: Correct. This matches our expanded and simplified expression.

6. Expand the following brackets:5(2f + 9w)

Explanations: To expand the brackets $5(2f + 9w)$, you need to distribute the number outside the parentheses, which is 5, to each term inside the parentheses. This means multiplying 5 by $2f$ and then by $9w$. - Multiplying 5 by $2f$ gives $10f$. - Multiplying 5 by $9w$ gives $45w$. So, combining these results, you get $10f + 45w$.

Option Analysis:

Option A: Incorrect. It incorrectly multiplies the first term inside the parentheses and does not distribute properly.
Option B: Incorrect. This is just the original expression, not expanded.
Option C: Incorrect. It only correctly multiplies the first term but misses the second term entirely.
Option D: Correct. It accurately represents the result of distributing 5 to both terms inside the parentheses.

7. $4x\left(x+2\right)-3x=$

Explanations: The expression $4x(x+2)-3x$ can be expanded by distributing the $4x$ across the terms inside the parentheses and then combining like terms. Step-by-step: - Distribute: $4x \cdot x + 4x \cdot 2 - 3x = 4x^2 + 8x - 3x$ - Combine like terms: $4x^2 + (8x - 3x) = 4x^2 + 5x$ Thus, the correct answer is D) $4x^2+5x$.

Option Analysis:

Option A: Incorrect. No like terms are combined to result in just a single term.
Option B: Incorrect. The quadratic term should remain as $4x^2$, not be simplified further.
Option C: Incorrect. There is no squared term, and the linear coefficient is incorrect.
Option D: Correct. This matches the expanded form after combining like terms.

8. Simplify the expression:$5(2x-3y)$

Explanations: To simplify the expression $5(2x-3y)$, you need to distribute the 5 across both terms inside the parentheses. This means multiplying 5 by each term individually: \[5 \times 2x = 10x\] and \[5 \times (-3y) = -15y.\] Therefore, combining these results gives: \[10x - 15y.\]

Option Analysis:

Option A: Incorrect. It incorrectly combines the terms into a single product.
Option B: Incorrect. It omits the term involving $y$.
Option C: Correct. This is the result of correctly distributing 5 across both terms inside the parentheses.
Option D: Incorrect. It incorrectly combines the terms, adding instead of subtracting the product of 5 and $-3y$.

9. MULTIPLY OUT THE BRACKET12 (-W-11V)

Explanations: To expand the bracket (-W-11V) , you need to distribute the negative sign across each term inside the brackets. Step 1: Distribute -1 to W and -11V: -1 * (-W) = +W -1 * (-11V) = +11V So, (-W-11V) becomes +W + 11V . However, the original expression has a negative sign in front of the bracket. Therefore, applying this negative sign to each term inside the brackets results in: - (+W) = -W - (+11V) = -11V Thus, the correct expansion is -W - 11V . Multiplying by 12 gives us -12W - 132V .

Option Analysis:

Option A: Incorrect. It changes the signs incorrectly.
Option B: Incorrect. It changes the signs incorrectly.
Option C: Incorrect. It changes the signs incorrectly.
Option D: Correct. This is the accurate expansion and multiplication of the bracket.

10. Expand (3x-1)(x + 5)

Explanations: To expand the brackets, we use the distributive property (also known as FOIL for binomials: First, Outer, Inner, Last). For $(3x-1)(x + 5)$: First: $3x \cdot x = 3x^2$ Outer: $3x \cdot 5 = 15x$ Inner: $-1 \cdot x = -x$ Last: $-1 \cdot 5 = -5$ Adding these together, we get: \[3x^2 + 15x - x - 5 = 3x^2 + 14x - 5\]

Option Analysis:

Option A: Incorrect. Missing the term $15x$ and $-x$.
Option B: Correct. Matches the expanded form.
Option C: Incorrect. Missing the term $15x$ and $-x$.
Option D: Incorrect. Extra term $+5$ from combining like terms incorrectly.

11. Expand and Simplify 2(x+2)+5(x+3)

Explanations: To expand and simplify $2(x+2)+5(x+3)$, follow these steps: - First, distribute the 2 across $(x+2)$: 2x + 4 . - Next, distribute the 5 across $(x+3)$: 5x + 15 . - Combine like terms: $2x + 5x = 7x$ and $4 + 15 = 19$. Thus, the simplified expression is $7x + 19$.

Option Analysis:

Option A: Incorrect. Does not match the combined like terms.
Option B: Incorrect. The coefficient of x is wrong.
Option C: Correct. Matches the simplified expression $7x + 19$.
Option D: Incorrect. The constant term is wrong.

12. Expand $2(5a-3b + 4)$

Explanations: To expand the expression $2(5a-3b + 4)$, you need to distribute the 2 across each term inside the parentheses. This means multiplying 2 by each of the terms: $2 \times 5a = 10a$, $2 \times -3b = -6b$, and $2 \times 4 = 8$. Therefore, the expanded form is $10a-6b + 8$.

Option Analysis:

Option A: Incorrect. It does not include the term with $5a$ multiplied by 2.
Option B: Incorrect. The constant term should be positive, not negative.
Option C: Incorrect. The terms involving $a$ and $b$ are correct but the constant term is wrong.
Option D: Correct. It accurately represents the expanded form of the given expression.

13. Expand and simplify(3 + g) (5-g)

Explanations: To expand and simplify $(3 + g)(5 - g)$, use the distributive property (also known as FOIL for binomials): First: $3 \times 5 = 15$ Outer: $3 \times (-g) = -3g$ Inner: $g \times 5 = 5g$ Last: $g \times (-g) = -g^2$ Combine these results: $15 - 3g + 5g - g^2$. Simplify by combining like terms: $15 + 2g - g^2$.

Option Analysis:

Option A: Incorrect, as it includes an extra positive term.
Option B: Incorrect, as the sign of the last term is wrong.
Option C: Correct, matches the simplified expression.
Option D: Incorrect, as it includes an extra positive term and a squared term in the wrong order.

14. Error Analysis:Spot the Mistake-Scenario B:A student expands $-2(x-5)$ $-2x-10$

Explanations: The correct answer is D) -2x + 10 . When expanding the expression $-2(x-5)$, you need to distribute the $-2$ across both terms inside the parentheses. This means multiplying $-2$ by $x$ and then by $-5$. The multiplication of $-2$ and $x$ gives $-2x$, and the multiplication of $-2$ and $-5$ results in $+10$. Therefore, the expression simplifies to $-2x + 10$.

Option Analysis:

Option A: Incorrect. It suggests adding instead of distributing the $-2$.
Option B: Incorrect. It omits the correct sign for $+10$.
Option C: Incorrect. It incorrectly distributes the $-2$ as a positive number.
Option D: Correct. It accurately represents the distribution of $-2$ across both terms inside the parentheses.

15. Error Analysis:A student expanded $5(x-4)$ $5x-4$

Explanations: The correct answer is B) $5x-20$. When expanding the expression $5(x-4)$, you must distribute the 5 to both terms inside the parentheses, resulting in $5 \cdot x - 5 \cdot 4 = 5x - 20$.

Option Analysis:

Option A: Incorrect. Adding 20 instead of subtracting it results in an incorrect expression.
Option B: Correct. This is the result of correctly distributing the 5 to both terms inside the parentheses.
Option C: Incorrect. Subtracting 4 from $5x$ does not account for multiplying -4 by 5.
Option D: Incorrect. Subtracting 20 is necessary, not subtracting a variable x.

16. Expand the following brackets:10(a + 2b + 3c)

Explanations: To expand the brackets $10(a + 2b + 3c)$, you need to distribute the 10 across each term inside the parentheses. This means multiplying 10 by $a$, then by $2b$, and finally by $3c$. - Multiplying 10 by $a$ gives $10a$. - Multiplying 10 by $2b$ (which is equivalent to $2 \times b$) results in $20b$. - Multiplying 10 by $3c$ (which is equivalent to $3 \times c$) results in $30c$. Therefore, the correct expansion is $10a + 20b + 30c$.

Option Analysis:

Option A: Incorrect. It does not distribute the 10 correctly to each term inside the parentheses.
Option B: Incorrect. It incorrectly multiplies $2b$ by 1 instead of 10.
Option C: Correct. It accurately distributes the 10 across all terms, resulting in $10a + 20b + 30c$.
Option D: Incorrect. It incorrectly multiplies $3c$ by 1 instead of 10.

17. Multiply out the brackets-5(m-7)

Explanations: To multiply out the brackets, you need to distribute the -5 across the terms inside the parentheses: -5(m-7) = -5 \times m + (-5) \times (-7) . This simplifies to -5m + 35 .

Option Analysis:

Option A: Incorrect. It incorrectly applies the multiplication, resulting in a negative sign for the second term.
Option B: Correct. This is the result of properly distributing -5 across both terms inside the parentheses.
Option C: Incorrect. The signs are applied incorrectly, leading to an incorrect result.
Option D: Incorrect. It reverses the sign on the second term after distribution.

18. Guided Practice:Step-by-Step Expansion. Apply your process to expand:$2(4a + 3(b-5))$

Explanations: To expand the expression $2(4a + 3(b-5))$, follow these steps: 1. Distribute the 2 to each term inside the parentheses: - First, distribute it to $4a$: $2 \times 4a = 8a$. - Next, distribute it to $3(b-5)$. This can be further broken down into two parts: - $2 \times 3b = 6b$ - $2 \times (-5) = -10$ 2. Combine all the terms: - The expression becomes $8a + 6b - 10$. Therefore, the correct answer is D) $8a + 6b-30$ (Note: There seems to be a discrepancy in the problem statement as the correct expansion should be $8a + 6b - 10$, not $-30$).

Option Analysis:

Option A: Incorrect because it does not include both terms from inside the parentheses.
Option B: Incorrect as it incorrectly adds a term that should be subtracted.
Option C: Incorrect due to missing and incorrect terms.
Option D: Close but incorrect in the constant term, should be $-10$.

19. Factorise 12x + 18

Explanations: The claimed correct answer is A) 6(2x + 3). This factorization is correct because both terms in the expression $12x + 18$ can be divided by 6, which is their greatest common factor (GCF). Dividing each term by 6 gives us $2x$ and $3$, respectively. Therefore, the expression simplifies to $6(2x + 3)$.

Option Analysis:

Option A: Correct. The GCF of 12 and 18 is 6, and dividing each term by 6 results in $2x + 3$.
Option B: Incorrect. Dividing by 2 does not yield integer coefficients for the terms inside the parentheses.
Option C: Incorrect. The GCF is not 3, and dividing each term by 3 results in non-integer coefficients.
Option D: Incorrect. Dividing by 3 does not yield integer coefficients for the terms inside the parentheses.

20. Expand (2x + 3)(x + 7)

Explanations: To expand the brackets, we use the distributive property (also known as FOIL for binomials: First, Outer, Inner, Last). For $(2x + 3)(x + 7)$: - First terms: $2x \cdot x = 2x^2$ - Outer terms: $2x \cdot 7 = 14x$ - Inner terms: $3 \cdot x = 3x$ - Last terms: $3 \cdot 7 = 21$ Adding these together, we get: \[2x^2 + 14x + 3x + 21 = 2x^2 + 17x + 21\]

Option Analysis:

Option A: Incorrect. The coefficient of $x$ is 17, not -17.
Option B: Incorrect. The middle term should be $+17x$, not $-10x$.
Option C: Correct. This matches the expanded form of the expression.
Option D: Incorrect. The coefficient of $x$ is 17, not -10.

21. Simplify:8(x+2)+4(x+1)

Explanations: To expand the brackets, distribute the numbers outside the parentheses to each term inside: For $8(x+2)$: - Distribute 8: $8 \cdot x + 8 \cdot 2 = 8x + 16$ For $4(x+1)$: - Distribute 4: $4 \cdot x + 4 \cdot 1 = 4x + 4$ Combine the results: $8x + 16 + 4x + 4 = 12x + 20$

Option Analysis:

Option A: Correct. $12x + 20$ is the simplified expression.
Option B: Incorrect. The constant term should be 20, not 16.
Option C: Incorrect. The coefficient of x should be 12, not 17.
Option D: Incorrect. Both the coefficient and constant terms are wrong.

22. Simplify by combining like terms. 8x + 4d + 7y + y + 3x =

Explanations: The correct answer is C) 11x + 4d + 8y=. To simplify the expression $8x + 4d + 7y + y + 3x$, we combine like terms: - Combine the x terms: $8x + 3x = 11x$ - The term with d remains as it is: $4d$ - Combine the y terms: $7y + y = 8y$ Thus, the simplified expression is $11x + 4d + 8y$.

Option Analysis:

Option A: Incorrect. Combines x and d incorrectly.
Option B: Incorrect. Only combines x terms but misses y and d terms.
Option C: Correct. Properly combines all like terms.
Option D: Incorrect. Random combination of variables without proper simplification.

23. Solve for $x$ $3(2x-4)=24$

Explanations: Expanding the brackets in the equation $3(2x-4)=24$ gives us $6x-12=24$. Adding 12 to both sides simplifies it to $6x=36$, and dividing by 6 yields $x=6$.

Option Analysis:

Option A: Incorrect. Plugging $x=4$ into the original equation does not satisfy it.
Option B: Correct. Plugging $x=6$ satisfies the equation, as shown by the step-by-step solution above.
Option C: Incorrect. Plugging $x=3$ into the original equation does not satisfy it.
Option D: Incorrect. Plugging $x=5$ into the original equation does not satisfy it.

24. Expand the following brackets:8(3w + 1)

Explanations: To expand the brackets $8(3w + 1)$, you need to distribute the number outside the parentheses, which is 8, to each term inside the parentheses. This means multiplying 8 by $3w$ and then by 1. - First, multiply 8 by $3w$: $8 \times 3w = 24w$. - Next, multiply 8 by 1: $8 \times 1 = 8$. Combining these results gives you the expression $24w + 8$, which is the correct answer.

Option Analysis:

Option A: Incorrect. Multiplying 8 by 3w and 1 separately does not result in 8w + 1.
Option B: Incorrect. The term $3w$ is not affected by the multiplication with 8, so it remains unchanged.
Option C: Incorrect. This option shows the original expression before expansion.
Option D: Correct. After distributing 8 to both terms inside the parentheses, you get $24w + 8$.

25. EXPAND THE BRACKET7 (X-2Y + 3)

Explanations: To expand the bracket, we distribute the 7 to each term inside the parentheses: $7 \times (X - 2Y + 3)$. This results in $7X - 14Y + 21$.

Option Analysis:

Option A: Incorrect. The terms should not change sign.
Option B: Correct. Matches the distribution of 7 to each term inside the parentheses.
Option C: Incorrect. The signs of all terms are incorrect.
Option D: Incorrect. The signs and order of terms are reversed.

26. What is the highest common factor of 12 and 18?

Explanations: The highest common factor (HCF) of two numbers is the largest number that divides both of them without leaving a remainder. For 12 and 18, their factors are: - Factors of 12: 1, 2, 3, 4, 6, 12 - Factors of 18: 1, 2, 3, 6, 9, 18 The largest number that appears in both lists is 6. Therefore, the HCF of 12 and 18 is 6.

Option Analysis:

Option A: 2 - Not the highest factor common to both numbers.
Option B: 18 - 18 is not a factor of 12.
Option C: 12 - 12 is not a factor of 18.
Option D: 6 - Correct, as it is the highest number that divides both 12 and 18 without leaving a remainder.

27. Expand and simplify the following quadratic expression ..... (x + 4)(x + 6)

Explanations: To expand and simplify the quadratic expression $(x + 4)(x + 6)$, we apply the distributive property (also known as FOIL: First, Outer, Inner, Last): - **First:** $x \cdot x = x^2$ - **Outer:** $x \cdot 6 = 6x$ - **Inner:** $4 \cdot x = 4x$ - **Last:** $4 \cdot 6 = 24$ Adding these together, we get: \[x^2 + 6x + 4x + 24 = x^2 + 10x + 24\] This matches Option A.

Option Analysis:

Option A: Correct. $x^2 + 10x + 24$ is the simplified form.
Option B: Incorrect. The coefficient of $x$ should be 10, not 10 (repeated).
Option C: Incorrect. The constant term should be 24, not 24 (repeated).
Option D: Incorrect. Both the coefficient of $x$ and the constant term are incorrect.

28. Expand and simplify(p + 5) (p-1)

Explanations: To expand and simplify $(p + 5)(p - 1)$, apply the distributive property (also known as FOIL: First, Outer, Inner, Last): First: $ p \cdot p = p^2 $ Outer: $ p \cdot (-1) = -p $ Inner: $ 5 \cdot p = 5p $ Last: $ 5 \cdot (-1) = -5 $ Combine these results: $ p^2 - p + 5p - 5 $ Simplify by combining like terms: $ p^2 + 4p - 5 $

Option Analysis:

Option A: Incorrect, as it does not match the simplified expression.
Option B: Correct, matches the simplified expression.
Option C: Incorrect, as it changes the sign of the linear term and constant term.
Option D: Incorrect, as it adds an extra $ p $ to the linear term and changes the constant term.

29. Guided Practice:Step-by-Step Expansion. Apply your process to expand:$3(2y + 5(y-2))$

Explanations: To expand the expression $3(2y + 5(y-2))$, follow these steps: 1. First, simplify inside the parentheses: $2y + 5(y - 2) = 2y + 5y - 10 = 7y - 10$. 2. Then, distribute the 3 across the simplified expression: $3(7y - 10) = 21y - 30$. Thus, the correct answer is B) $21y-30$ .

Option Analysis:

Option A: Incorrect because it does not correctly distribute the 3 across all terms inside the parentheses.
Option B: Correct as shown by the step-by-step expansion process.
Option C: Incorrect due to an error in simplifying inside the parentheses and distributing the 3.
Option D: Incorrect for similar reasons as Option C, with additional errors in combining like terms.

30. Expand the expression:$7(3m-n)$

Explanations: To expand the expression $7(3m-n)$, you distribute the 7 to both terms inside the parentheses: \[7 \times 3m - 7 \times n = 21m - 7n\]

Option Analysis:

Option A: Correct. $21m-7n$ is the result of distributing 7 to both terms.
Option B: Incorrect. It does not follow the distributive property correctly.
Option C: Incorrect. Missing the negative sign before $7n$.
Option D: Incorrect. The term $-7n$ is missing, and there's an extra minus sign between terms.

Frequently Asked Questions

What is expanding brackets in algebra?

Expanding brackets involves removing the brackets from an expression by multiplying each term inside the brackets with the term outside. This process helps simplify expressions and solve equations.

Can expanding brackets be used in solving linear equations?

Yes, expanding brackets can be a crucial step in solving linear equations. It helps to simplify the equation and make it easier to isolate the variable.

How do you handle negative coefficients when expanding brackets?

When a bracket has a negative coefficient, each term inside the bracket is multiplied by that negative number. This changes the sign of each term within the bracket.

What role does factorization play in expanding brackets?

Factorization can be used to simplify expressions before or after expanding brackets. It helps identify common factors that can be factored out, making the expression easier to work with.

What is the highest common factor (HCF) in the context of expanding brackets?

The highest common factor (HCF) refers to the largest number that divides two or more terms without leaving a remainder. In algebra, it can be used to simplify expressions by factoring out the HCF before or after expanding brackets.

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