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Chapter IV review exercises
Textbook: pp. 125–128
GoalConsolidate the factoring methods of Chapter IV in mixed exercises.
New words
factorization · ko‘paytuvchilarga ajratishshort multiplication formulas · qisqa ko‘paytirish formulalarigrouping · guruhlashroots · ildizlar
Explanation
In Chapter IV we learned four tools for factoring: taking out a common factor, grouping, the formulas for (a ± b)², and the formulas for a² - b² and cubes. The order is: common factor first, then a formula, then grouping. Factoring not only simplifies expressions but also helps solve equations: if a product equals zero, at least one factor equals zero. For example 3x² - 12x = 0 gives 3x(x - 4) = 0, so x = 0 or x = 4. Formulas also help in computing: 75² - 25² = (75 - 25)(75 + 25) = 5000. Always check the result by multiplying.
Worked examples
x²y - y³ = y(x² - y²) = y(x - y)(x + y).
9 - (a - 2)² = (3 - (a - 2))(3 + (a - 2)) = (5 - a)(1 + a).
Class activity
“Factoring relay”: team members factor a polynomial in turn: the first takes out the common factor, the second applies a formula, the third checks.
Practice
1
Solve 3x² - 12x = 0 by factoring.
x = 0 or x = 4
2
Compute 75² - 25² with a formula.
5000
3
Factor a²b - 2ab² + b³ completely.
b(a - b)²
4
What can be said if a product equals zero?
At least one of the factors must equal zero.