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Lesson #53 Notes
53 FACTORING THE DIFFERENCES OF TWO SQUARES AND THE GENERAL TRINOMIAL p. 676-678, p.680-684 Factoring the Differences of Two Squares p.676-678 An expression in the form of a2 - b2 is called the difference of two squares. Factoring such an expression is the reverse of multiplying the sum of two terms by the difference of the same two terms. Therefore: a2 - b2 = (a + b)(a - b) Remember that for a monomial to be a square: 1. Its numerical coefficient must be a square. 2. The exponent of each of its variables must be an even number. Procedure: Express each of the binomials terms as the square of a monomial; then apply the rule: a2 - b2 = (a + b)(a - b) example: x2 - 25 = x2 - 52 = (x + 5)(x - 5) Model Problems p. 677 - Go over all 4 examples. Factoring Trinomials of the Form: ax2 + bx + c p.680-684 Factoring a trinomial of the form ax2 + bx + c is the reverse of multiplying binomials of the form (dx + e)(fx + g). When factoring a trinomial like this, list the possible pairs of factors using combinations of factors of the first and last terms and test them, one by one, until the correct middle term is found. example: x2 + 5x + 6

1. The product of the first terms must be x2. Therefore: (x )(x ) 2. Since the product of the last terms must be +6, the last terms must be either positive or negative. Therefore, possibilities are: (+1)(+6) (+2)(+3) (-1)(-6) (-2)(-3) 3. Since the sum of these factors must equal +5, the only set that fits this criteria is: (+2)(+3) 4. Therefore the factors of x2 + 5x + 6 must be: (x + 2)(x + 3) 5. Test this conclusion by multiplying the two binomial factors. The result must be the given trinomial. (x + 2)(x + 3) = x2 + 5x + 6 Procedure: 1. The product of the first terms of the binomials must be equal to the first term in the trinomial (ax2). 2. The product of the last terms of the binomial must be equal to the last term of the trinomial (c). 3. When the first term of each binomial is multiplied by the second term of the other and the sum of these products is found, this result must equal the middle term of the trinomial (bx). Remember: < If the last term is positive, the last terms of the binomials must be either both positive or both negative. < If the last term is negative, the last terms of binomials must be one positive and negative. Model Problems p. 682-684 - examples: y2 - 8y + 12 = (y - 6)(y - 2) or (y - 2)(y - 6) c2 + 5c - 6 = (c + 6)(c - 1) or (c - 1)(c + 6) 2a2 - 7a - 15 = (2a + 3)(a - 5)

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