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Techniques of Factoring Polynomials

dmaye avatar
dmaye
36 diapositives · 396 Views · Jan 05, 2020

Learn various techniques to factor polynomials including GCF factoring, factoring trinomials, box method, and difference of two squares. Practice examples provided for better understanding.

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Techniques of Factoring Polynomials
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  1. MODULE 14 Review of Factoring

  2. Factoring Polynomials This process is basically the REVERSE of the distributive property. distributive property factoring

  3. In factoring you start with a polynomial (2 or more terms) and you want to rewrite it as a product(or as a single term) Three terms One term Factoring Polynomials

  4. Factor out the GCF:  Techniques of Factoring Polynomials • Factoring Out the Greatest Common Factor(GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial.

  5. Write the two terms in the form of prime factors… They have in common 2yy Factoring Polynomials - GCF This process is basically the reverse of the distributive property.

  6. Check the work….

  7. 3 terms 2 - 3a c +2b c 2 b 2 2 One term Factoring Polynomials - GCF Factor the GCF: a 4 ( ) b

  8. Factoring Polynomials - GCF EXAMPLE: 5x • 3

  9. ExamplesFactor the following polynomial.

  10. ExamplesFactor the following polynomial.

  11. Techniques of Factoring Polynomials 2.Factoring a Polynomial with four or more Terms by Grouping There is no GCF for all four terms. In this problem we factor GCF by grouping the first two terms and the last two terms.

  12. Example 1 Factor 4m + 8n +mp + 2np = (4m + 8n) + (mp + 2np) p (m + 2n) + (m + 2n) = 4 (4 + p) = (m + 2n)

  13. Example 2 Factor ax – 5ay + 2x – 10y = (ax – 5ay) + (2x – 10y) (x – 5y) (x – 5y) + 2 = a (a + 2) = (x – 5y)

  14. Example 3 Factor ab + 3a – 3b – 9 = (ab + 3a) + (– 3b – 9) (– 3) (b + 3) (b + 3) + = a = (b + 3) (a – 3)

  15. We need to find factors of 6 Techniques of Factoring Polynomials 3. Factoring Trinomials. ….that add up to5 Since 6 can be written as the product of 2 and 3 and 2 + 3 = 5, we can use the numbers 2 and 3 to factor the trinomial.

  16. 2x3 = 6 2 + 3 = 5 Factoring Trinomials, continued... Use the numbers 2 and 3 to factor the trinomial… Write the parenthesis, with An “x” in front of each. Write in the two numbers we found above.

  17. Factoring Trinomials, continued... So we factored the trinomial… You can check your work by multiplying back to get the original answer

  18. 6 and 1 – 6 and 1 3 and –2 Find factors of 6 that add up to 7 Factoring Trinomials Find factors of –6 that add up to –5 Find factors of –6 that add up to 1

  19. Factoring Trinomials factors of 6 that add up to 7: 6 and 1 factors of –6 that add up to – 5: – 6and 1 factors of –6 that add up to 1:3 and – 2

  20. First: Multiply 2 and –6: 2 (– 6) = – 12 – 3 and 4 Factoring TrinomialsThe hard case – “Box Method” Note: The coefficient of x2 is different from 1. In this case it is 2 1 Next: Find factors of – 12that add up to 1

  21. Factoring TrinomialsThe hard case – “Box Method” • Draw a 2 by 2 grid. • Write the first term in the upper left-hand corner • Write the last term in the lower right-hand corner.

  22. Find factors of – 12that add up to 1 Factoring TrinomialsThe hard case – “Box Method” – 3x 4= – 12 – 3+4= 1 • Take the two numbers–3and4, and put them, complete • with signs and variables, in the diagonal corners, like this: x –3 It does not matter which way you do the diagonal entries! x 4

  23. The hard case – “Box Method” • Then factor like this: Factor Top Row Factor Bottom Row From Left Column From Right Column

  24. The hard case – “Box Method” Note:The signs for the bottom row entry and the right column entry come from the closest term that you are factoring from. DO NOT FORGET THE SIGNS!! + + Now that we have factored our box we can read off our answer:

  25. The hard case – “Box Method” – 16 and – 3 Look for factors of 48 that add up to –19 Finally, you can check your work by multiplying back to get the original answer.

  26. Use“Box” method to factor the following trinomials. • 2x2+ 7x + 3 • 2. 4x2– 8x – 21 • 3. 2x2– x – 6

  27. Check your answers. • 2x2+ 7x + 3 = (2x + 1)(x + 3) • 2. 2x2– x – 6 = (2x + 3)(x– 2) • 3. 4x2– 8x – 21 = (2x– 7)(2x + 3)

  28. Note… • Not every quadratic expression can be • factored into two factors. • For example x2– 7x + 13. • We may easily see that there are no factors of 13 that added up give us –7 • x2– 7x + 13 is a prime trinomial.

  29. a2 – b2 = (a + b)(a – b) FORMULA: Factoring the Difference of Two Squares a2– ab + ab –b2 = a2 – b2 (a + b)(a – b) = The difference of the squares of two quantitiesalways factors into the sum of these two quantities multiplied by the difference of these two quantities.

  30. 2 2 (5) (4r) Difference Of two squares 2 2 (x) Difference of two squares (2y) Factoring the difference of two squares a2 – b2 = (a + b)(a – b) Factor x2 – 4y2 Factor 16r2 – 25 (x – 2y)(x + 2y) (4r – 5)(4r + 5) Now you can check the results…

  31. More Examples 2 2 2 (y – 4)(y + 4) y – 16 = (y) (4) = - 2 2 2 (5x) – (9) 25x – 81 = (5x – 9)(5x +9) = 2 2 2 4 2 x – 100y (x ) – (10y) = 2 2 (x – 10y)(x +10y) =

  32. Factoring the Sum Two Cubes Using the following formula:

  33. The following polynomial can be factored: (n + 3) = Example 1 = (2n + 5) = = (2n + 5)

  34. Example 2 Factor = = =

  35. Factoring the Difference of Two Cubes Using the following formula: The following polynomial can be factored: (n – 3) =

  36. Example 1 Factor = = Example 2 Factor = =