Factoring Special Products MATH 017 Intermediate Algebra S

Factoring Special Products MATH 017 Intermediate Algebra S. Rook

Overview • Section 5. 7 in the textbook – Factor perfect square trinomials – Factor difference & sum of two squares – Factor difference & sum of two cubes – Factor completely 2

Factor Perfect Square Trinomials

Factor Perfect Square Trinomials • Remember to ALWAYS look for a GCF before factoring! • Consider (a + b)2 = a 2 + 2 ab + b 2 • a 2 comes from squaring a in (a + b)2 • 2 ab comes from doubling the product of a and b in (a + b)2 • b 2 comes from squaring b in (a + b)2 4

Factor Perfect Square Trinomials (Example) Ex 1: Factor 49 x 2 + 84 x + 36 5

Factor Perfect Square Trinomials (Example) Ex 2: Factor 27 x 2 - 72 x + 48 6

Factor Differences & Sums of Two Squares

Difference of Two Squares • Remember to ALWAYS look for a GCF before factoring! • Consider (a + b)(a – b) • (a + b)(a – b) = a 2 – b 2 • a 2 comes from the F term in (a + b)(a – b) • b 2 comes from the L term in (a + b)(a – b) 8

Sum of Two Squares • Remember to ALWAYS look for a GCF before factoring! • Consider a 2 + b 2 • Only 3 possibilities for the factoring: (a + b) = a 2 + 2 ab + b 2 ≠ a 2 + b 2 (a – b) = a 2 – 2 ab + b 2 ≠ a 2 + b 2 (a + b)(a – b) = (a – b)(a + b) = a 2 – b 2 ≠ a 2 + b 2 • Therefore, the sum of two squares is PRIME 9

Differences & Sums of Two Squares (Example) Ex 3: Factor 7 x 2 – 63 10

Differences & Sums of Two Squares (Example) Ex 4: Factor 4 x 2 + 49 11

Differences & Sums of Two Squares (Example) Ex 5: Factor 9 x 2 – 16 12

Factor Differences & Sums of Two Cubes

Sum & Difference of Two Cubes • Remember to ALWAYS look for a GCF before factoring! • Consider (a + b)(a 2 – ab + b 2) a(a 2 – ab + b 2) + b(a 2 – ab + b 2) a 3 – a 2 b + ab 2 + a 2 b – ab 2 + b 3 a 3 + b 3 • Consider (a – b)(a 2 + ab + b 2) a(a 2 + ab + b 2) – b(a 2 + ab + b 2) a 3 – b 3 14

Sum & Difference of Two Cubes • Thus: a 3 + b 3 = (a + b)(a 2 – ab + b 2) a 3 – b 3 = (a – b)(a 2 + ab + b 2) a 3 (+/ –) b 3 = (a b)(a ab + b 2) |__same__| | |__opposite____| 15

Sum & Difference of Two Cubes (Example) Ex 6: Factor 8 w 3 – 27 z 3 16

Sum & Difference of Two Cubes (Example) Ex 7: Factor 5 x 3 + 320 y 3 17

Factoring Completely

Factoring Completely • Remember to ALWAYS look for a GCF before factoring! • Factor as normal • Look at the result to see if any of the products can be factored further – Polynomials with a degree of 1 or less cannot be factored further • e. g. 2 x + 1, 7 – The trinomial from factoring a sum or difference of cubes cannot be factored further 19

Factoring Completely (Example) Ex 8: Factor completely: 2 p 4 – 2 20

Factoring Completely (Example) Ex 8: Factor completely: m 4 n – 256 n 21

Summary • After studying these slides, you should know how to do the following: – Be able to RECOGNIZE and COMPLETELY factor: • Perfect square trinomials • Sum & Difference of two squares • Sum & Difference of two cubes 22