ACT Preparation / Unit 8 ACT 0706 worked example, 60 to 90 seconds
Expanding a Product
The idea
To expand a product, work in four steps. First, check for a special pattern, a square of a binomial or a sum times a difference, and use its formula. Second, otherwise, multiply every term of the first factor by every term of the second, keeping each sign with its term. Third, collect like terms by power. Fourth, check by substituting a small number, such as x = 1, into both the original and the result.
In practice
Worked example: expand (2x − 3)(x² + 4x − 1). Step two: 2x times each term gives 2x³ + 8x² − 2x; −3 times each gives −3x² − 12x + 3. Step three: 2x³ + 5x² − 14x + 3. Step four: at x = 1, the original is (−1)(4) = −4, and the result is 2 + 5 − 14 + 3 = −4. They match.
Where people slip
One matching value does not prove the expansion, since different polynomials can agree at a point. A second value makes the check stronger: at x = 2, both give 11.
Your place on the trail
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