Expansion preserves the whole expression
Every term in one factor multiplies every term in the other. Coefficients include signs, and absent powers have coefficient zero
- Four products Multiply each term in the first binomial by both terms in the second. Keep the signs
- Collect like powers Add signed coefficients only when the variable and power match \(\displaystyle 8x-3x=5x\)
- Square a binomial Keep both cross-products. Subtract twice the product of the two terms, and add both squares \(\displaystyle (a-b)^2=a^2-2ab+b^2\)
- Sum times difference Matching first terms and opposite second terms cancel their cross-products \(\displaystyle (a+b)(a-b)=a^2-b^2\)
Worked example
Expand (3y − 2)(2y + 5), keeping all four products
- \(\displaystyle 3y\times2y=6y^2\)First term by first term
- \(\displaystyle 3y\times5=15y\)First term by second term
- \(\displaystyle (-2)\times2y=-4y\)Second term by first term
- \(\displaystyle (-2)\times5=-10\)Second term by second term
- \(\displaystyle 15y-4y=11y\)Collect only the linear terms
- \(\displaystyle 6y^2+11y-10\)The complete expanded form. The identity holds for every real y
Common mistakes
Difficulty levels
Questions about this topic
Does every pair of brackets need the same shortcut?+
No. Multiplying every term always works. A shortcut is useful only when the full bracket pattern matches
Why is the final term of a difference squared positive?+
The subtracted term multiplies itself to give a square. For example, in (x − 4)² the last product is (−4) × (−4) = 16; the cross-products add to −8x
What do I enter for a missing power?+
Enter zero for its coefficient. For example, the coefficient of x in 9x² − 4 is zero
Can an area diagram use negative lengths?+
No. Its lengths must be positive. Algebraic expansion still works for negative variable values even when they no longer describe that physical rectangle
Is one substitution enough to prove an identity?+
No. It can expose an error, but agreement at one value is not proof. Expansion must preserve every coefficient
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 8: Algebraic expressions and fractions