Factoring Polynomials

1. Factoring Polynomials This process is basically the REVERSE of the distributive property. DISTRIBUTIVE PROPERTY: In factoring you start with a polynomial (2 or more terms)and you want to rewrite it as a product (or a single term)

2. Factoring Polynomials • Greatest Common Factor (GCF) The GCF for a polynomial is the largest monomial that divides (is a factor of ) each term of the polynomial. Factor out the GCF:  This process is basically the reverse of the distributive property.

3. Factoring Polynomials - GCF 2 terms-written in the form of prime factors… They have in common 2yy

4. Factoring Polynomials - GCF EXAMPLE:

5. Factoring Polynomials - GCF EXAMPLE:

6. More examples:Factor the following polynomials • 12x2 - 20x4 = (4x2×3) – (4x2×5x2) • = 4x2( 3 – 5x2 ) Both have 4x2 in common • 15x3 y5 + 3x2 y4 = (3x2 y4×5xy)+ (3x2 y4×1) • = 3x2 y4 (5xy + 1) Both have 3x2y4 in common Remember to include 1

7. Factoring Polynomials Factoring a Polynomial with Four Terms by Grouping. There is no GCF for all four terms. We factor by grouping the first two terms and the last two terms.

8. Factoring Polynomials Factoring Trinomials. We need to find factors of 6 that add up to 5 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.

9. Factoring Polynomials Factoring Trinomials, continued... 2x3 = 6 2 + 3 = 5 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.

10. Factoring Polynomials Factoring Trinomials, continued... Check your work by multiplying back to get the original answer

11. Factoring Polynomials Factoring Trinomials Find factors of 6 that add up to 7 Find factors of 6 that add up to –5 Find factors of 6 that add up to 1

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

13. 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

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

15. Factoring TrinomialsThe hard case – “Box Method” – 3x 4 = – 12 – 3+4 = 1 Find factors of – 12that add up to 1 • Draw a 2 by 2 grid, and write the first term in the upper • left-hand corner and the last term in the lower right-hand corner.

16. Factoring TrinomialsThe hard case – “Box Method” – 3x 4 = – 12 – 3+4 = 1 • Take the two numbers –3 and 4, and put them, complete • with signs and variables, in the diagonal corners, like this It does not matter which way you do the diagonal entries.

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

18. 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: Finally, you can check your work by multiplying back to get the original answer.

19. 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.

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

21. 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)

22. 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.

23. Factoring the Difference of Two Squares FORMULA a2 – b2 = (a + b)(a – b) The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.

24. Factoring the difference of two squares Factor x2 – 4y2 = x2 – (2y)2 (change to the difference of two squares) = (x – 2y)(x + 2y) Factor 16r2 – 25 = (4r)2 – (5)2 (change to the difference of two squares) = (2r – 5)(2r + 5)

25. *Fits the form of a diff. of two squares Difference of two squares

26. *Fits the form of a diff. of two squares Difference of two squares

27. *Fits the form of a diff. of two squares so we need to continue factoring… Difference of two squares