Perfect square trinomials: Level 2 Problem 8. Without multiplying out a) explain why (1 − x)² = (x − 1)². Because (1 − x) is the negative of (x − 1). And (−a)² = a² for any quantity a. b) explain why (1 − x)³ = −(x − 1)³. (−a)³ = −a³ for any quantity a. * The following problems show how we can go from what we know to what we do not know. Problem 9. Use your knowledge of (a + b)² to multiply out (a + b)3. [Hint: (a + b)3 = (a + b)(a + b)²] To see the answer, pass your mouse over the colored area.
Problem 10. Multiply out (x + 2)3.
Problem 11. Multiply out (x − 1)3.
Problem 12. The square of a trinomial. Use your knowledge of [Hint: Treat as a binomial with as the first term.] Show that it will equal the sum of the squares of each term, plus twice the product of all combinations of the terms.
Problem 13. Can you generalize the result of the previous problem? Can you immediately write down the square of (a + b + c + d)?
Completing the square x² + 8x + _?_ = (x + _?_)² When the coefficient of x² is 1, as in this case, then to complete a perfect square trinomial, we must add a square number. What square number must we add? We must add the square of half of coefficient of x. The trinomial will then be the square of x plus half that coefficient. x² + 8x + 16 = (x + 4)² We add the square of half the coefficient of x -- which in this case is 4 -- because when we multiply (x + 4)², the coefficient of x will be twice that number. Problem 14.
(Skill in Arithmetic, Lesson 26.) Example 7. Complete the square: x² − 7x + ? = (x − ?)²
And since the middle term of the trinomial has a minus sign, then the binomial also must have a minus sign. Problem 15. Complete the square. The trinomial is the square of what binomial? a) x² + 4x + ? x² + 4x + 4 = (x + 2)² b) x² − 2x + ? x² − 2x + 1 = (x − 1)² c) x² + 6x + ? x² + 6x + 9 = (x + 3)² d) x² − 10x + ? x² − 10x + 25 = (x − 5)² e) x² + 20x + ? x² + 20x + 100 = (x + 10)²
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