Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Algebraic expressions and exponent laws

Algebraic expressions and exponent laws

Expand, factorise and simplify algebraic expressions, including powers and polynomial forms.

Updated 2026-07-26 - 11 min read

Algebraic expressions and exponent laws is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Expand algebraic products

Use the distributive property so every term in one factor multiplies every term in the other. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by expand algebraic products and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Both cross-products are required: (x + a)(x + b) = x² + (a + b)x + ab.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Expand (x + 3)(x + 5).

Step 1 - identify the governing idea: Use the distributive property so every term in one factor multiplies every term in the other.

Step 2 - apply it to this evidence: The products are x², 5x, 3x and 15.

Result: x² + 8x + 15

The relationship is visible in the working: The products are x², 5x, 3x and 15. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x² + 15 — It does not agree with the required relationship: The products are x², 5x, 3x and 15.
  • x² + 8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2x² + 15 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Expand (2x − 1)(x + 4).

Step 1 - identify the governing idea: Use the distributive property so every term in one factor multiplies every term in the other.

Step 2 - apply it to this evidence: Distribution gives 2x² + 8x − x − 4.

Result: 2x² + 7x − 4

The relationship is visible in the working: Distribution gives 2x² + 8x − x − 4. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2x² + 8x − 4 — It does not agree with the required relationship: Distribution gives 2x² + 8x − x − 4.
  • 2x² + 3x − 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2x² + 7x + 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Simplify 3x(x − 2) + x².

Step 1 - identify the governing idea: Use the distributive property so every term in one factor multiplies every term in the other.

Step 2 - apply it to this evidence: Expansion gives 3x² − 6x, then add x².

Result: 4x² − 6x

The relationship is visible in the working: Expansion gives 3x² − 6x, then add x². Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 4x − 6 — It does not agree with the required relationship: Expansion gives 3x² − 6x, then add x².
  • 3x² − 5x — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4x³ − 6x — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Factorise algebraic expressions

Reverse expansion by taking common factors or finding factors whose sum and product match a quadratic. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by factorise algebraic expressions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Every term must be represented inside the factorised form, which can be verified by expansion.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Factorise 6x² − 9x.

Step 1 - identify the governing idea: Reverse expansion by taking common factors or finding factors whose sum and product match a quadratic.

Step 2 - apply it to this evidence: The greatest common factor is 3x.

Result: 3x(2x − 3)

The relationship is visible in the working: The greatest common factor is 3x. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 3(2x² − 3x) — It does not agree with the required relationship: The greatest common factor is 3x.
  • x(6x − 9x) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 6x(x − 9) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Factorise x² + 7x + 12.

Step 1 - identify the governing idea: Reverse expansion by taking common factors or finding factors whose sum and product match a quadratic.

Step 2 - apply it to this evidence: The factors 3 and 4 multiply to 12 and add to 7.

Result: (x + 3)(x + 4)

The relationship is visible in the working: The factors 3 and 4 multiply to 12 and add to 7. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • (x + 6)(x + 2) — It does not agree with the required relationship: The factors 3 and 4 multiply to 12 and add to 7.
  • (x − 3)(x − 4) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (x + 12)(x + 1) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Factorise x² − 25.

Step 1 - identify the governing idea: Reverse expansion by taking common factors or finding factors whose sum and product match a quadratic.

Step 2 - apply it to this evidence: This is a difference of two squares.

Result: (x − 5)(x + 5)

The relationship is visible in the working: This is a difference of two squares. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • (x − 5)² — It does not agree with the required relationship: This is a difference of two squares.
  • (x − 25)(x + 1) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x(x − 25) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Apply exponent laws to algebraic expressions

For matching non-zero bases, products add exponents, quotients subtract exponents and powers multiply exponents. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by apply exponent laws to algebraic expressions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Index laws describe multiplication and division of powers, not sums such as x² + x³.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Simplify x⁴ × x⁷.

Step 1 - identify the governing idea: For matching non-zero bases, products add exponents, quotients subtract exponents and powers multiply exponents.

Step 2 - apply it to this evidence: Matching bases multiply by adding exponents.

Result: x¹¹

The relationship is visible in the working: Matching bases multiply by adding exponents. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x²⁸ — It does not agree with the required relationship: Matching bases multiply by adding exponents.
  • 2x¹¹ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Simplify (3a²b)³.

Step 1 - identify the governing idea: For matching non-zero bases, products add exponents, quotients subtract exponents and powers multiply exponents.

Step 2 - apply it to this evidence: Cube the coefficient and multiply each exponent by 3.

Result: 27a⁶b³

The relationship is visible in the working: Cube the coefficient and multiply each exponent by 3. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 9a⁵b³ — It does not agree with the required relationship: Cube the coefficient and multiply each exponent by 3.
  • 27a⁵b — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3a⁶b³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Simplify 12m⁸ ÷ 3m³.

Step 1 - identify the governing idea: For matching non-zero bases, products add exponents, quotients subtract exponents and powers multiply exponents.

Step 2 - apply it to this evidence: Divide coefficients and subtract exponents: 8 − 3 = 5.

Result: 4m⁵

The relationship is visible in the working: Divide coefficients and subtract exponents: 8 − 3 = 5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 9m⁵ — It does not agree with the required relationship: Divide coefficients and subtract exponents: 8 − 3 = 5.
  • 4m²⁴ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4m¹¹ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Expand (2x − 1)(x + 4).
  2. Factorise x² + 7x + 12.
  3. Simplify (3a²b)³.

Answers

  1. 2x² + 7x − 4 — Distribution gives 2x² + 8x − x − 4.
  2. (x + 3)(x + 4) — The factors 3 and 4 multiply to 12 and add to 7.
  3. 27a⁶b³ — Cube the coefficient and multiply each exponent by 3.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources