1. Beginning Algebra 5.6 Factoring: A General Review

2. Objective 1. To factor a variety of polynomials. A General Review 6.5 Factoring: A General Review

3. 1. Check for common factors factors and factor them out out of the polynomial. no common factor 2. If there is no common factor check for the two special types factorable polynomials special types of factorable polynomials:. Difference of Squaresbinomial  a) Difference of Squares (binomial). Perfect Square Trinomial  b) Perfect Square Trinomial factors of general polynomial 3. Check for factors of general polynomial.

4. Check for common factors factors and factor them out out of the polynomial. Given a general polynomial polynomial: What to do How to do it 25ax 3 - 40abx 2 + 15a 2 x 2 5ax 2 (5x - 8b + 3a) Factor Common Factors: Polynomials Given a general binomial binomial: 64a 2 x 2 + 16b 2 x 2 16x 2 (4a 2 + b 2 ) prime factor Check for common factors factors and factor them out out of the polynomial.

5. What to do How to do it Factor general polynomials:  Given a special binomial 64a 2 x 2 – 16b 2 x 2 16x 2 (4a 2 – b2)b2) difference of squares Check for common factors factors and factor them out out of the polynomial. (2a) 2 – (b) 2 difference of squares prime factors (2a + b)(2a – b) Rewrite square factors factor(s) in the inner binomial binomial. Factor as sum sum and difference of bases of the squares squares. 64a 2 x 2 – 16b 2 x 2 = 16x 2 (2a + b)(2a – b) Multiply by common factor factor 16x 2

6. What to DoHow to Do It Given Even Degree Binomial Binomial: (ab) 2 – (3c) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (ab + 3c)(ab – 3c) a 2 b 2 – 9c 2 = (ab + 3c)(ab – 3c) Rewrite square factors factor(s) in each term. Factor as sum sum and difference of bases of the squares squares. The factors factors of difference difference of squares form the product product of conjugate pairs pairs. a 2 b 2 – 9c 2

7. What to DoHow to Do It 3x 2 – 12 3(x 2 – 4) difference of squares prime factors Factor binomial as the difference of squares Multiply by common factor factor 3 left in composite or power form. Difference of Squares Factor Difference of Squares: (x + 2)(x – 2) 3x 2 – 12 = 3(x + 2)(x – 2) Factor out the common factor(s) from each term. Given Second Degree Binomial Binomial:

8. What to DoHow to Do It Given Even Degree Binomial Binomial: x 4 – y 8 (x 2 ) 2 – (y 4 ) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (x 2 + y 4 )(x + y 2 )(x – y2)y2) x 4 – y 8 = (x 2 + y 4 )(x + y 2 )(x – y2)y2) Rewrite square factors factor(s) in each term. Factor second binomial binomial again. Completely factor the difference difference of squares squares requires repeated factoring factoring. Factor as sum sum and difference of bases of the squares squares. (x 2 + y 4 )(x 2 – y4)y4) difference of squares x 2 – y 4

9. What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: a 2 x 2  2abx + b 2 = (ax  b) 2 (ax) 2  2abx + b2b2 twice ax twice b 2abx Perfect square trinomials must have the first first and last terms terms be perfect squares and the last sign positive positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign  is the sign of the binomial binomial. a 2 x 2  2abx + b 2 (ax  b) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms.

10. What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: 25x 2 + 60x + 36 = (5x + 6) 2 (5x) 2 + 60x + 6262 Perfect square trinomials trinomials must have the first first and last terms terms be perfect squares and the last sign positive positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign + is the sign of the binomial binomial. 25x 2 + 60x + 36 (5x + 6) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms. twice 5x twice 6 2  5x  6 60x 60x

11. What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: 16x 2 – 56x + 49 = (4x – 7) 2 (4x) 2 – 56x + 7272 Perfect square trinomials trinomials must have the first first and last terms terms be perfect squares and the last sign positive positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign – is the sign of the binomial binomial. 16x 2 – 56x + 49 (4x – 7) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms. twice 4x twice 7 2  4x  7 56x 56x

12. What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: 81x 2 – 90x + 25 = (9x – 5) 2 (9x) 2 – 90x + 5252 Perfect square trinomials trinomials must have the first first and last terms terms be perfect squares and the last sign positive positive. If all of these conditions hold, check to see if the middle term is twice twice the product product of square roots roots of first term term and last term term. The middle sign sign – is the sign of the binomial binomial. 81x 2 – 90x + 25 (9x – 5) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms. twice 9x twice 5 2  9x  5 90x 90x

13. Find the product product of the first and last coefficients coefficients: AA AA and C Find all of the pairs of factors r r and s Given a general quadratic trinomial trinomial: What to do How to do it is the middle coefficient. rs = B r – s = B or r + s = B so their sum P = r  s s, r > s Ax 2  Bx  C AC A  C = P Factor general quadratic trinomial: difference or difference is the middle coefficient.

14. Clue of Signs Factor by Clue of Signs: What to DoHow to Do It Ax 2  Bx  C Read  as “+ or or ”””” Given general trinomial that has no common factor. See clues of the signs signs. P = ACAC ACAC is the grouping number (r (r – s) = B whose sum sum or difference difference is B + sum Find all possible factors of GN = P P = rs rs, r > s (r (r + s) = B difference – difference Sum + Difference Same sign  Larger sign  Rewrite middle term term Bx Bx : rx sx Ax 2  rx  sx + C underline underline and factor by grouping (ax  b)(cx  d)

15. Find all of the pairs pairs of factors: r and s Find the product product of 6 6 and 15 15 : Middle sign sign is – – therefore: 90 · 1 GN: 6 ·15 = 90 – 10, + 9 What to do How to do it 10 – 9 = 1 18 · 5 15 · 6 30 · 3 45 · 2 Separate middle term term 1t–1·t:1t–1·t: – 10t, + 9t with the difference difference = 1.1. 10 · 9 90 Example: 6t 2 – t – 15

16. Copy the polynomial polynomial: Rewrite middle term term –– –– tt:tt: and group for factoring Factor each group group: Factor common factor factor. –10t + 9t– 6t 2 – 10t + 9t – 15 6t 2 –t –– –– 15 What to do How to do it – (2t + 3)(3t – 5) Continue Example: 6t 2 – t – 15 2t(3t –– –– 5) + 3(3t – – 5)5) Always Check Factors. – (2t + 3)(3t – 5) See next slide: + bring down middle sign underline common factor:

17. Check Factors using FOIL 6t 2 – t – 15 = (2t + 3)(3t – 5) What to DoHow to Do It  Check: Check: Multiply using First Note sum of O + I terms F 0 I L (2t + 3)(3t - 5) Outer Inner Last 6t 2 –– –– 10t + 9t –– –– 15 6t 2 - 15 - 10t + 9t Always Check Factors. 6t 2 –– –– 1t –– –– 15

18. What to DoHow to Do It Ax 2  Bx  C k·(ax 2  bx  c) k·ax 2  k·bx  k·ck·c Factor Factor out the common s factor(s) from each term term. Apply the distributive property property. Common Factors Trinomials with Common Factors: As common factors factors numbers are left in composite form form and letters letters are left in power form form. Check Inner Polynomial for Clue of Signs Signs and GN ax 2  bx  c

19. What to DoHow to Do It –– 72x 2 – 60x – 28 4(18x 2 – 15x – 7)7) 4·18x 2 – 4·15x – 4·74·7 Factor out the common s factor(s) from each term. distributive property Apply the distributive property. Common Factors Trinomials with Common Factors: Numbers as common factors are left in composite form form. Check: Inner Polynomial for Clue of Signs Signs and GN 18x 2 – 15x – 7 Hold the 4

20. Inner Trinomial18x 2 - 15x - 7 Inner Trinomial: 18x 2 - 15x - 7 Find all of the pairs of factors: r and s Find the product product of 18 18 and 77:77: Middle sign is – – therefore: 18 · 7 = 126 - 21, + 6 What to do How to do it Separate middle term term - 15x 15x : - 21x, + 6x with the difference difference = 15. 126 · 1 21 - 6 = 15 21 · 6 18 · 7 42 · 3 63 · 2 14 · 9 126

21. Copy the polynomial polynomial: Rewrite middle term term -15x : and group for factoring Factor each group group: Factor common factor factor. – 21x + 6x– 18x 2 – 21x + 6x – 7 18x 2 –– –– 15x –– –– 7 What to do How to do it – (3x + 1)(6x – 7) –– Continue Example: 18x 2 – 15x – 7 3x(6x - 7) 7) + 1(6x - 7)7) Always Check Factors. – (3x + 1)(6x – 7) See next slide: + bring down middle sign underline common factor:

22. Check Factors by FOIL What to DoHow to Do It  Check factors of inner trinomial trinomial by First Find the sum of O + I terms F 0 I L – (3x + 1)(6x – 7) –– 18x 2 – 15x – 7 Outer Inner Last 18x 2 –– –– 21x + 6x –– –– 7 18x 2 - - 7 - - 21x + 6x Multiply by common factor factor 4 –– 72x 2 – 60x – 28 = 4(3x + 1)(6x –– –– 7)

23. Given a quadratic trinomial and integers a, b, c The test requires knowing squares of integers or use to a calculator. What to do How to do it Review squares of integers ax 2 + bx + c Test for Factorability Quadratic Trinomial  Test for Factorability: The trinomial will factor with have rational factors factors if: b 2 –– –– 4ac = d2d2 a > 0 signs included in b and c d  0d  0d  0d  0 b 2 –– –– 4ac = perfect square trinomial 0 perfect square trinomial. factorable trinomial d 2 factorable trinomial. d2d2 = +d

24. What to do How to do it  Test 4x 2 –– –– 12x - 94x 2 –– –– 12x –– –– 9 – (-12) 2 – 4(4)(-9) = 288 not factorable a = 4, b = -12, c = -9 4x 2 –– –– 12x - 9 = (2a –– –– 3) 2 b2b2 b2b2 –– –– 4ac = 0  d 2  Test 4x 2 –– –– 12x + 94x 2 –– –– 12x + 9 – (-12) 2 – 4(4)(9) perfect square trinomial a = 4, b = -12, c = 9 = 0 Perfect square trinomial yields square of binomial Test for Factorability Quadratic Trinomial  Test for Factorability:

25. What to do How to do it  Test 4x 2 +15x + 94x 2 + 15x + 9 – (-15) 2 – 4(4)(9) = 81 factorable trinomial a = 4, b = -15, c = 9 4x 2 + 15x + 9 = (4x + 3)(x + 3) Sum: Sum: 12 + 3 = 15 = 9 2  Factor Factor 4x 2 + 15x + 9 4x 2 + 15x + 9 4x(x + 3) + 3(x + 3) GN = 36 36 -- factors factors 12 and 3 Complete Complete the factors: Test for Factorability Quadratic Trinomial  Test for Factorability: 4x 2 + 12x + 3x + 9

26. Perfect square trinomial trinomial. Factor: common factors factors: What to do How to do it ax + ay = a(x + y) a2x2 a2x2  2abx + b2b2 Factor Polynomials Summary: Factor Polynomials Difference of Squares Squares:A 2 –– –– B 2 = (A + B)(A –– –– B) (ax  b) 2 Distribute: left / rightax + bx = (a + b)x Conjugate Pairs Pairs: a 2 x 2 –– –– b2y2 b2y2 = binomial squared (ax+by)(ax –– –– by)

27. Difference of of like even even powers What to do How to do it xn xn - y n = (x 2m –– –– y 2m ) (x m + y m )(x m –– –– ym)ym) Factor Polynomials Summary: Factor Polynomials Repeat since m = 4, p = 2 (x m + y m ) is prime (x 4 +y 4 )(x 2 + y 2 )(x 2 –– –– y2)y2) (x 4 + y 4 )(x 4 –– –– y4)y4) [(x 2 ) 2 –– –– (y 2 ) 2 ] n = 4, m = 2:2: Repeat since m = 2 (x m + y m ) is prime – (x 2 + y 2 )(x 2 – y 2 ) (x 2 + y 2 )(x + y)( x –– –– y) n = 8, m = 4, p = 2:2: x4 x4 – y 4 = x 8 – y 8 = Repeat if m = 2p (x m + y m ) is prime (x 4 +y 4 )(x 2 + y 2 )(x+ y)(x –– –– y)

28. Factor by grouping grouping. What to do How to do it Apply commutative property Factor Polynomials Summary: Factor Polynomials ax + by - bx - ay (a b)(x y) = (a – b)(x – y) Polynomials Polynomials of 4or more terms rearranged / grouped grouped: = ax – ay – bx + by = a(x – y) – b(x – y) Apply distributive property: Underline common factor: Apply distributive property = a(x – y) – b(x – y) Underline groups Underline groups:

29. THE END 5.6