Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §5.5 Factor Special Forms Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

5.4 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §5.4 → Factoring TriNomials Any QUESTIONS About HomeWork §5.4 → HW-14

§5.5 Factoring Special Forms Factoring Perfect-Square Trinomials and Differences of Squares Recognizing Perfect-Square Trinomials Factoring Perfect-Square Trinomials Recognizing Differences of Squares Factoring Differences of Squares Factoring SUM of Two Cubes Factoring DIFFERENCE of Two Cubes

Recognizing Perfect-Sq Trinoms A trinomial that is the square of a binomial is called a perfect-square trinomial A2 + 2AB + B2 = (A + B)2 A2 − 2AB + B2 = (A − B)2 Reading the right sides first, we see that these equations can be used to factor perfect-square trinomials. A2 + 2AB + B2 = (A + B)(A + B) A2 − 2AB + B2 = (A − B)(A − B)

Recognizing Perfect-Sq Trinoms Note that in order for the trinomial to be the square of a binomial, it must have the following: 1. Two terms, A2 and B2, must be squares, such as: 9, x2, 100y2, 25w2 2. Neither A2 or B2 is being SUBTRACTED. 3. The remaining term is either 2  A  B or −2  A  B where A & B are the square roots of A2 & B2

Example  Trinom Sqs Determine whether each of the following is a perfect-square trinomial. a) x2 + 8x + 16 b) t2 − 9t − 36 c) 25x2 + 4 – 20x SOLUTION a) x2 + 8x + 16 Two terms, x2 and 16, are squares. Neither x2 or 16 is being subtracted. The remaining term, 8x, is 2x4, where x and 4 are the square roots of x2 and 16

Example  Trinom Sqs SOLUTION b) t2 – 9t – 36 Two terms, t2 and 36, are squares. But 36 is being subtracted so t2 – 9t – 36 is not a perfect-square trinomial SOLUTION c) 25x2 + 4 – 20x It helps to write it in descending order. 25x2 – 20x + 4

Example  Trinom Sqs SOLUTION c) 25x2 − 20x + 4 Two terms, 25x2 and 4, are squares. There is no minus sign before 25x2 or 4. Twice the product of the square roots is 2  5x  2, is 20x, the opposite of the remaining term, −20x Thus 25x2 − 20x + 4 is a perfect-square trinomial.

Factoring a Perfect-Square Trinomial The Two Types of Perfect-Squares A2 + 2AB + B2 = (A + B)2 A2 − 2AB + B2 = (A − B)2

Example  Factor Perf. Sqs Factor: a) x2 + 8x + 16 b) 25x2 − 20x + 4 SOLUTION a) x2 + 8x + 16 = x2 + 2  x  4 + 42 = (x + 4)2 A2 + 2 A B + B2 = (A + B)2

Example  Factor Perf. Sqs Factor: a) x2 + 8x + 16 b) 25x2 − 20x + 4 SOLUTION b) 25x2 – 20x + 4 = (5x)2 – 2  5x  2 + 22 = (5x – 2)2 A2 – 2 A B + B2 = (A – B)2

Example  Factor 16a2 – 24ab + 9b2 SOLUTION 16a2 − 24ab + 9b2 = (4a)2 − 2(4a)(3b) + (3b)2 = (4a − 3b)2 = (4a − 3b)(4a − 3b) CHECK: (4a − 3b)(4a − 3b) = 16a2 − 24ab + 9b2  The factorization is (4a − 3b)2.

Expl  Factor 12a3 – 108a2 + 243a SOLUTION Always look for a common factor. This time there is one. Factor out 3a. 12a3 − 108a2 + 243a = 3a(4a2 − 36a + 81) = 3a[(2a)2 − 2(2a)(9) + 92] = 3a(2a − 9)2 The factorization is 3a(2a − 9)2

Recognizing Differences of Squares An expression, like 25x2 − 36, that can be written in the form A2 − B2 is called a difference of squares. Note that for a binomial to be a difference of squares, it must have the following. There must be two expressions, both squares, such as: 9, x2, 100y2, 36y8 The terms in the binomial must have different signs.

Difference of 2-Squares Diff of 2 Sqs → A2 − B2 Note that in order for a term to be a square, its coefficient must be a perfect square and the power(s) of the variable(s) must be even. For Example 25x4 − 36 25 = 52 The Power on x is even at 4 → x4 = (x2)2 Also, in this case 36 = 62

Example  Test Diff of 2Sqs Determine whether each of the following is a difference of squares. a) 16x2 − 25 b) 36 − y5 c) −x12 + 49 SOLUTION a) 16x2 − 25 The 1st expression is a sq: 16x2 = (4x)2 The 2nd expression is a sq: 25 = 52 The terms have different signs. Thus, 16x2 − 25 is a difference of squares, (4x)2 − 52

Example  Test Diff of 2Sqs SOLUTION b) 36 − y5 The expression y5 is not a square. Thus, 36 − y5 is not a diff of squares SOLUTION c) −x12 + 49 The expressions x12 and 49 are squares: x12 = (x6)2 and 49 = 72 The terms have different signs. Thus, −x12 + 49 is a diff of sqs, 72 − (x6)2

Factoring Diff of 2 Squares A2 − B2 = (A + B)(A − B) The Gray Area by Square Subtraction The Gray Area by (LENGTH)(WIDTH)

Example  Factor Diff of Sqs Factor: a) x2 − 9 b) y2 − 16w2 SOLUTION a) x2 − 9 = x2 – 32 = (x + 3)(x − 3) A2 − B2 = (A + B)(A − B) b) y2 − 16w2 = y2 − (4w)2 = (y + 4w)(y − 4w) A2 − B2 = (A + B) (A − B)

Example  Factor Diff of Sqs Factor: c) 25 − 36a12 d) 98x2 − 8x8 SOLUTION c) 25 − 36a12 = 52 − (6a6)2 = (5 + 6a6)(5 − 6a6) d) 98x2 − 8x8 Always look for a common factor. This time there is one, 2x2: 98x2 − 8x8 = 2x2(49 − 4x6) = 2x2[(72 − (2x3)2] = 2x2(7 + 2x3)(7 − 2x3)

Grouping to Expose Diff of Sqs Sometimes a Clever Grouping will reveal a Perfect-Sq TriNomial next to another Squared Term Example Factor m2 − 4b4 + 14m + 49  rearranging  m2 + 14m + 49 − 4b4  GROUPING  (m2 + 14m + 49) − 4b4

Grouping to Expose Diff of Sqs Example Factor m2 − 4b4 + 14m + 49 Recognize m2 + 14m + 49 as Perfect Square Trinomial → (m+7)2 Also Recognize 4b4 as a Sq → (2b)2 (m2 + 14m + 49) − 4b4  Perfect Sqs  (m + 7)2 − (2b2)2 In Diff-of-Sqs Formula: A→m+7; B→2b2

Grouping to Expose Diff of Sqs Example Factor m2 − 4b4 + 14m + 49 (m + 7)2 − (2b2)2  Diff-of-Sqs → (A − B)(A + B)  ([m+7] − 2b2)([m + 7] + 2b2)  Simplify → ReArrange  (−2b2 + m + 7)(2b2 + m + 7) The Check is Left for us to do Later

Factoring Two Cubes The principle of patterns applies to the sum and difference of two CUBES. Those patterns SUM of Cubes DIFFERENCE of Cubes

TwoCubes SIGN Significance Carefully note the Sum/Diff of Two-Cubes Sign Pattern SAME Sign OPP Sign SAME Sign OPP Sign

Example: Factor x3 + 64 Factor Recognize Pattern as Sum of CUBES Determine Values that were CUBED Map Values to Formula Substitute into Formula Simplify and CleanUp

Example: Factor 8w3−27z3 Factor Recognize Pattern as Difference of CUBES Determine CUBED Values Simplify by Properties of Exponents Map Values to Formula Sub into Formula Simplify & CleanUp

 Example: Check 8w3−27z3 Check Use Distributive property Use Comm & Assoc. properties, and Adding-to-Zero 

Sum & Difference Summary Difference of Two SQUARES SUM of Two CUBES Difference of Two CUBES

Factoring Completely Sometimes, a complete factorization requires two or more steps. Factoring is complete when no factor can be factored further. Example: Factor 5x4 − 3125 May have the Difference-of-2sqs TWICE

Factoring Completely SOLUTION 5x4 − 3125 = 5(x4 − 625) = 5(x − 5)(x + 5)(x2 + 25) The factorization: 5(x − 5)(x + 5)(x2 + 25)

Factoring Tips Always look first for a common factor. If there is one, factor it out. Be alert for perfect-square trinomials and for binomials that are differences of squares. Once recognized, they can be factored without trial and error. Always factor completely. Check by multiplying.

WhiteBoard Work Problems From §5.5 Exercise Set 14, 22, 48, 74, 94, 110 The SUM (Σ) & DIFFERENCE (Δ) of Two Cubes

All Done for Today Sum of Two Cubes

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

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