ALGEBRAIC OPERATIONS

QUESTION 1 [4 marks] — Criterion A Medium

Simplify each product:

a. $3x \times 4y$

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$$12xy$$

b. $5a \times (-2a)$

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$$-10a^2$$

c. $(3x)^2$

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$$9x^2$$

d. $2p \times 3q \times 4p$

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$$24p^2q$$

QUESTION 2 [4 marks] — Criterion A Medium

A rectangle has width $4x$ and length $7x$.

a. Write and simplify an expression for the area of the rectangle.

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$$\text{Area} = 4x \times 7x = 28x^2$$

b. If $x=3$ cm, find the numerical area, and verify it matches substituting $x=3$ directly into your simplified expression.

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Width $=12$ cm, length $=21$ cm, area $=12\times21=252$ cm$^2$. Check via formula: $28(3)^2=28\times9=252$ cm$^2$ ?.

QUESTION 3 [4 marks] — Criterion B Medium

Investigate the pattern when multiplying powers of the same variable, e.g. $x^2 \times x^3$.

a. Expand $x^2 \times x^3$ fully (writing out all the individual $x$ factors) to find the simplified result.

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$x^2\times x^3 = (x\times x)\times(x\times x\times x) = x\times x\times x\times x\times x = x^5$

b. Repeat for $x^3\times x^4$ and $x^2\times x^5$, and state the pattern connecting the original exponents to the final exponent.

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$x^3\times x^4=x^7$. $x^2\times x^5=x^7$. Pattern: when multiplying powers of the same base, ADD the exponents: $x^m\times x^n=x^{m+n}$.

QUESTION 4 [4 marks] — Criterion C Medium

A student simplifies $(2x)^2$ as $2x^2$.

a. Explain the student's error, and give the correct simplification.

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$(2x)^2$ means $(2x)\times(2x)$, where BOTH the 2 and the $x$ get squared: $=4x^2$, not $2x^2$. The student forgot that the exponent applies to everything inside the brackets, including the coefficient 2.

b. Verify the difference by evaluating both $(2x)^2$ and $2x^2$ at $x=3$.

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$(2x)^2=(6)^2=36$. $2x^2=2(9)=18$. These are different (36 vs 18), confirming the error.

QUESTION 5 [4 marks] — Criterion D Medium

A square sheet of metal has side length $5x$ cm. A smaller square with side length $2x$ cm is cut from one corner.

a. Write and simplify an expression for the area of the original large square.

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$$(5x)^2=25x^2 \text{ cm}^2$$

b. Write and simplify an expression for the area of the small square cut out.

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$$(2x)^2=4x^2 \text{ cm}^2$$

c. Write a simplified expression for the remaining area of metal, and evaluate it when $x=4$ cm.

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Remaining area $=25x^2-4x^2=21x^2$. At $x=4$: $21(16)=336$ cm$^2$.

QUESTION 6 [6 marks] — Criterion A Hard

Simplify each product:

a. $(2x)^3$

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$$8x^3$$

b. $3x^2 \times 4x^3$

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$$12x^5$$

c. $(-3x)^2 \times 2x$

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$$9x^2 \times 2x = 18x^3$$

QUESTION 7 [8 marks] — Criterion B Hard

Investigate the pattern in $(x+a)^2-(x-a)^2$ for different values of $a$.

a. Fully expand $(x+5)^2-(x-5)^2$.

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$(x^2+10x+25)-(x^2-10x+25) = 20x$.

b. Repeat with $(x+3)^2-(x-3)^2$ and $(x+7)^2-(x-7)^2$.

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$(x+3)^2-(x-3)^2=12x$. $(x+7)^2-(x-7)^2=28x$.

c. State the general pattern: what is $(x+a)^2-(x-a)^2$ in terms of $x$ and $a$? Verify your general rule using one of your specific results.

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$(x+a)^2-(x-a)^2 = 4ax$. Check with $a=5$: $4(5)x=20x$ ?, matching part (a).

QUESTION 8 [5 marks] — Criterion B Hard

Investigate whether $(x+y)^3$ equals $x^3+y^3$ (i.e. whether cubing 'distributes' over addition, the way some students assume).

a. Test with $x=2, y=3$: calculate $(x+y)^3$ and separately $x^3+y^3$.

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$(2+3)^3=5^3=125$. $2^3+3^3=8+27=35$. These are very different ($125\ne35$).

b. Fully expand $(x+y)^3=(x+y)(x+y)(x+y)$ step by step to find the CORRECT general expansion, confirming it does NOT simplify to just $x^3+y^3$.

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$(x+y)(x+y)=x^2+2xy+y^2$. Then $(x^2+2xy+y^2)(x+y) = x^3+x^2y+2x^2y+2xy^2+xy^2+y^3 = x^3+3x^2y+3xy^2+y^3$. This is clearly NOT just $x^3+y^3$ — it has two extra 'cross terms' ($3x^2y$ and $3xy^2$) that a naive 'distribute the power' approach would miss.

QUESTION 9 [5 marks] — Criterion C Hard

A student simplifies $2x \times 3x$ as $6x$ (adding the coefficients of $x$ together, treating multiplication like addition).

a. Explain the student's error, distinguishing between how coefficients behave under MULTIPLICATION versus ADDITION of like terms.

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The student is confusing two different rules: for ADDITION of like terms ($2x+3x$), you DO add the coefficients, giving $5x$. But for MULTIPLICATION ($2x\times3x$), you multiply the coefficients (2×3=6) AND apply the index law for the variable part ($x\times x=x^2$), giving $6x^2$, not $6x$.

b. Verify the correct answer ($6x^2$) using $x=2$, comparing to the student's incorrect $6x$.

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$2x\times3x$ at $x=2$: $4\times6=24$. Correct formula $6x^2$ at $x=2$: $6(4)=24$ ?. Student's $6x$ at $x=2$: $12\ne24$ — confirms the error.

QUESTION 10 [5 marks] — Criterion C Hard

A classmate says: '$(x^2)^3$ and $x^2\times x^3$ both involve the numbers 2 and 3 with $x$, so they must be equal.'

a. Calculate both expressions using the correct index laws, and determine whether the classmate is right.

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$(x^2)^3=x^{2\times3}=x^6$ (power of a power: MULTIPLY exponents). $x^2\times x^3=x^{2+3}=x^5$ (product of powers: ADD exponents). These are NOT equal ($x^6\ne x^5$), so the classmate is wrong.

b. Verify numerically at $x=2$: calculate $(2^2)^3$ and $2^2\times2^3$ directly.

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$(2^2)^3=4^3=64$. $2^2\times2^3=4\times8=32$. Different ($64\ne32$), confirming the two expressions are genuinely different.

QUESTION 11 [6 marks] — Criterion D Hard

A rectangular garden bed's length is $3x$ metres and width is $2x$ metres. A second, similar bed has BOTH dimensions doubled (length $6x$, width $4x$).

a. Find the area of each bed, in terms of $x$.

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Bed 1: $3x\times2x=6x^2$. Bed 2: $6x\times4x=24x^2$.

b. Find the RATIO of Bed 2's area to Bed 1's area, and explain why doubling BOTH dimensions doesn't just double the area (connect this to why the ratio isn't simply 2).

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Ratio $=\frac{24x^2}{6x^2}=4$. Doubling both dimensions doesn't just double the area because area depends on the PRODUCT of two dimensions — doubling each of two multiplied quantities multiplies the result by $2\times2=4$, not just 2. This is a general principle: scaling both dimensions of a 2D shape by a factor $k$ scales the area by $k^2$.

Algebraic products