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Factoring

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Factoring

1. 1. <ul><li>Factoring is the process of finding polynomials, called factors , whose product equals a given polynomial. </li></ul><ul><li>A polynomial that cannot be written as a product of polynomials with integer coefficients is a prime or irreducible polynomial . </li></ul><ul><li>A polynomial is factored completely when it is written as a product of prime polynomials with integer coefficients. </li></ul>
2. 2. <ul><li>Example Factor out the greatest common factor. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) </li></ul><ul><li>(b) </li></ul>
3. 3. <ul><li>Example Factor each polynomial by grouping. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) </li></ul>
4. 4. <ul><li>Solution </li></ul><ul><li>(b) </li></ul>
5. 5. <ul><li>Example Factor each trinomial. </li></ul><ul><li>(a) </li></ul><ul><li>(b) </li></ul>
6. 6. <ul><li>Solution (a) We must find integers a , b , c , d such that </li></ul><ul><li>By FOIL, we see that ac = 4 and bd = 6. Thus a and c </li></ul><ul><li>are 1 and 4 or 2 and 2. </li></ul><ul><li>Since the middle term is negative, consider only </li></ul><ul><li>negative values for b and d . The possibilities are –1 </li></ul><ul><li>and –6 or –2 and –3. </li></ul>
7. 7. <ul><li>Solution (a) Try these combinations of factors </li></ul><ul><li>The last trial gives the correct factorization. </li></ul>
8. 8. <ul><li>Solution (b) Try the various combinations of factors </li></ul><ul><li>The last trial gives the correct factorization. </li></ul>
9. 9. Perfect Square Trinomials
10. 10. <ul><li>Example Factor each polynomial. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) </li></ul><ul><li>(b) </li></ul>
11. 11. <ul><li>Solution </li></ul>
12. 12. Difference of Squares
13. 13. <ul><li>Example Factor each polynomial. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) </li></ul><ul><li>(b) </li></ul>
14. 14. Difference and Sum of Cubes Difference of cubes Sum of cubes
15. 15. <ul><li>Example Factor each polynomial. </li></ul><ul><li>(a) (b) </li></ul><ul><li>Solution </li></ul><ul><li>(a) </li></ul>
16. 16. <ul><li>Solution </li></ul><ul><li>(b) </li></ul>
17. 17. <ul><li>Example Factor the polynomial </li></ul><ul><li> </li></ul><ul><li>Solution Replacing 2 a – 1 with m and factoring gives </li></ul><ul><li>Now, replace m with 2 a – 1 in the factored form and </li></ul><ul><li>simplify. </li></ul>