Department of Mathematics

Pindling Basic Algebra   by Example Series

Key Concepts: Understand factoring of polynomials.

Skills to Learn

1. Know how to factor common monomial

2. Know how to factor by grouping

3. Know how to factor the difference of two squares

4. Know how to factor trinomials

5. Know how to factor trinomials by grouping

6. Know how to factor the sum and difference of two cubes

7. Know some general techniques for factoring

Factor of Monomials

Example 1. Factor Note: is common to all monomials So Factor out 2xy 2xy(x + 3y + 2) So

Example 2. Factor -3a2bc + 15a3b2c - 9a2bc2 Note -3a2 bc is a common term in each monomial So factor out -3a2 bc -3a2bc(1 - 5ab + 3c) Note: is often proper when factoring to do so that the first term factored has a positive (+) sign

Factor By Grouping

Example 3. Factor ax + ay + bx + by ax + ay + bx + by (ax + ay) + (bx + by) a(x + y) + b(x + y) = (a + b)(x + y)

Example 4. Factor x2 - 4x + xy - 4y (x2 - 4x) + (xy - 4y) x(x - 4) + y(x - 4) (x + y)(x - 4)

Factoring the Difference of Two Squares

Example 5. Factor This is a perfect square (12x + 10)(12x - 10)	Example 6. Factor This is a perfect square (3m + n)(3m - n)
Example 7. Factor out 2 to get (4x2 - 16) is a perfect square 2(2x + 4)(2x - 4)	Example 8. Factor out 5 to get (x2 - 25) is a perfect square 5(x + 5)(x - 5)

Factoring Trinomials

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Factoring Trinomials

Example 9. Factor x2 + 2x -15 Note: So

Example 10. Factor 6x2 - 7x + 2 Note: Factor for 2 is ; Since +2 and -7 Either (a) (b) No solution in either Or Note: Factor for 2 is ; Since +2 and -7 Either (a) The Solution (b) , So

Example 11. Factor Factors of 10: 10 x 1 or 2 x 5 Factors of -6: 6 x 1 or 2 x 3 ( ) Possible solutions are: (a) (b) (c) (d) (e) (f) or Solution (g) (h)

Example 12. Factor  8x3 + 2x2 - 3x   Factor out x: x(8x2 + 2x - 3) 8x3 + 2x2 - 3x   (factors of 8: 8 x 1 or 2 x 4) Factors of 3 is 1 x 3 ( ) Possible solutions are: (a) (b) (c) or x Sol (d) So  8x3 + 2x2 - 3x  = x(2x - 1)(4x + 3)

Factoring Cubes and Other Factoring Strategies

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Examples 13. Factor Trinomial So	Examples 14. Factor Trinomial or So
Example 15. Factor Factor difference of 2 squares like , where a = x+2, b =y So	Example 16. Factor by grouping Let So Since Then