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Factoring

Factoring . Page 80: Factoring GCF. 3x 2 + 18x + 9 4k 4 + 18k 3 – 6k 2 3X 2 + 6X + 3. =3(x 2 + 6x + 3). =2k 2 (2k 2 + 9k – 3). Mistake on the homework page 80 #14) answer is P the problem is messed up. =3(X 2 + 2X + 1) . Page 97: Factor by Grouping. X 2 + 3X + XK + 3K

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Factoring

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  1. Factoring

  2. Page 80: Factoring GCF • 3x2 + 18x + 9 • 4k4 + 18k3 – 6k2 • 3X2 + 6X + 3 =3(x2 + 6x + 3) =2k2(2k2 + 9k – 3) Mistake on the homework page 80 #14) answer is P the problem is messed up =3(X2 + 2X + 1)

  3. Page 97: Factor by Grouping • X2 + 3X + XK + 3K • M2 + MN – 3M – 3N =X(X + 3) + K(X + 3) =(X + K) (X + 3) or (X + 3) (X + K) =M(M + N) – 3(M + N) =(M – 3) (M + N) or (M + N) (M – 3)

  4. Page 84 top left: Multiply • (a + 5) (a – 5) • (a2 – 6) (a2 + 6) = a2 – 5a + 5a – 25 = a2 – 25 = a4 + 6a2 – 6a2 – 36 = a4 – 36

  5. Page 84: Factoring Difference of Squares • 36x2 – 121y2 • 4a16 – 225 • What about x2 + 25? Does it factor? = (6x – 11y) (6x + 11y) = (2a8 – 15) (2a8 + 15) Try (x + 5) (x + 5) or (x + 5) (x – 5) or (x – 5) (x – 5) Conclusion: Sum of Squares is NOT factorable. We call it prime.

  6. Page 90: Quick Factor • N2 + 6N + 5 • N2 + 2N – 15 • A2 – 29AB + 100B2 • A2 – 14AB – 32B2 Factors of +5 Sum of +6 = (N + 5) (N + 1) Factors of -15 Diff of +2 = (N – 3) (N + 5) 1*15 3*5 Factors of +100 Sum of -29 = (A – 4B) (A – 25B) 1*100 2*50 4*25 10*10 Factors of -32 Diff of -14 = (A + 2B) (A – 16B) 1*32 2*16 4*8

  7. Factoring: Sum & Difference of Cubes • A3 – B3 = (A – B)(A2 + AB + B2) • A3 + B3 = (A + B)(A2 – AB + B2) • 1000X3 + 27 • 64X3 – 1 = (10X + 3) (100X2 – 30X + 9) = (4X – 1) (16X2 + 4X + 1)

  8. Page 91: Long Factor • 3x2 + 7x + 2 • 8u2 – 9u + 1 • 7n2 – 13n – 2 Factors of +6 Sum of +7 = 3x2 + 1x + 6x + 2 1*6 2*3 =x(3x + 1) + 2(3x + 1) = (x + 2)(3x + 1) Factors of +8 Sum of -9 = 8u2 – 1u – 8u + 1 1*8 2*4 = (u – 1)(8u – 1) =u(8u – 1) – 1(8u – 1) Factors of -14 Diff of -13 = 7n2 + 1n – 14n – 2 1*14 2*7 = (n – 2)(7n + 1) =n(7n + 1) – 2(7n + 1)

  9. Page 93: Factoring, All Mixed Up • Binomials: • Take out GCF • Check for Diff of Squares, Diff/Sum of Cubes • Trinomials: • Take out GCF • Check for Quick Factor (coef in front = 1) • Check for Long Factor (coef in front ≠ 1)

  10. Page 94: All Mixed Up with GCF • 3x2 – 15x + 18 • 60m3 + 54m2 – 6m = 3(x2 – 5x + 6) Factors of +6 Sum of -5 1*6 2*3 = 3(x – 2) (x – 3) = 6m(10m2 – 9m – 1) Factors of -10 Diff of -9 1*10 2*5 = 6m(10m2 + 1m – 10m – 1) = 6m[ m(10m + 1) – 1(10m + 1) ] = 6m (m – 1) (10m + 1)

  11. Page 93: Factoring, All Mixed Up • Binomials: • Take out GCF • Check for Diff of Squares & Diff/Sum of Cubes • A2 – B2 = (A – B)(A +B) • A3 – B3 = (A – B)(A2 + AB + B2) • A3 + B3 = (A + B)(A2 – AB + B2) • Trinomials: • Take out GCF • Check for Quick Factor (coef in front = 1) • Check for Long Factor (coef in front ≠ 1)

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