using the Diamond or "ac" method Factoring Factoring Trinomials using the Diamond or "ac" method 5.4a S KM & PP

Whew! We finally “guessed” correctly! Factoring We could “GUESS and CHECK” and hope to find binomials that multiply to . Whew! We finally “guessed” correctly! Guess FOIL to check 5.4a S KM & PP

Now let’s learn how to find those FOIL numbers! Factoring It would have been easier if we knew the FOIL terms for the trinomial so that we could factor by grouping. Have a look! Now let’s learn how to find those FOIL numbers! 5.4a S KM & PP

What’s the Diamond? Example 1 The “diamond” is a number puzzle that can help us find the FOIL numbers for a trinomial. Look at a few examples and see if you can discover the pattern! ? ? 5.4a S KM & PP

What’s the Diamond? Example 2 Try this one! ? ? 5.4a S KM & PP

What’s the Diamond? Example 3 Have you figured it out? ? ? 5.4a S KM & PP

What’s the Diamond? Example 4 Here it is in ALGEBRA terms! ? ? 5.4a S KM & PP

Can You Solve This Diamond? Which numbers add to 11 and multiply to 30? ? ? Factor pairs of 30 1 30 2 15 3 10 5 6 5.4a S KM & PP

Which numbers add to 13 and multiply to 36? How about Another? Which numbers add to 13 and multiply to 36? ? ? Factor pairs of 36 1 36 2 18 3 12 4 9 6 5.4a S KM & PP

Which numbers add to -10 and multiply to -24? Think about + signs! Which numbers add to -10 and multiply to -24? ? ? Factor pairs of 24 1 24 2 12 3 8 4 6 5.4a S KM & PP

ax2 + bx + c Why the Diamond? ? ? The coefficients a, b, and c of ax2 + bx + c are used in the diamond as follows: ? ? This is sometimes called the “ac” method. 5.4a S KM & PP

Factor Using the Diamond: Example 1 2x2 - 11x - 40 First use the diamond to find the FOIL numbers: a = 2 b = -11 c = -40 ac = 2(-40) = -80 Factor 80 1 80 2 40 4 20 5 16 8 10 5.4a S KM & PP

Now We Can Factor by Grouping 2x2 - 11x - 40 Now rewrite the middle term using the FOIL numbers and factor by grouping! 5.4a S KM & PP

Or we could use the Generic Rectangle! or by the commutative property 5.4a S KM & PP

Factor Using the Diamond: Example 2 6x2 - 17x +12 First use the diamond to find the FOIL numbers: a = 6 b = -17 c = 12 ac = 6(12) = 72 Factor 72 1 72 2 36 3 24 4 18 6 12 8 9 5.4a S KM & PP

Now using Grouping Example 2 6x2 - 17x +12 Now rewrite the middle term using the FOIL numbers and factor by grouping! 5.4a S KM & PP

Factor Using the Diamond: Example 3 6x2 - 13xy – 28y2 First use the diamond to find the FOIL numbers: a = 6 b = -13 c = -28 Factor 168 1 168 2 84 3 56 4 42 6 28 7 24 8 21 12 14 We can stop our list at or at 12 5.4a S KM & PP

Now factor by grouping! 6x2 - 13xy – 28y2 Now rewrite the middle term using the FOIL numbers and factor by grouping! 5.4a S KM & PP

Factor with the Diamond: Example 4: x2 + 12x + 20 a = 1 b = 12 c = 20 1 20 2 10 4 5 x2 + 12x + 20 = x2 + 2x +10x + 20 = x(x + 2) + 10(x + 2) = (x + 2) (x + 10) 5.4a S KM & PP

Factor with the Diamond: Example 5: x2 + x - 12 a = 1 b = 1 c = -12 1 12 2 6 3 4 x2 + x - 12 = x2 - 3x + 4x - 12 = x(x - 3) + 4(x - 3) = (x - 3) (x + 4) 5.4a S KM & PP

Factor with the Diamond: Example 6: x2 + 8x + 15 a = 1 b = 8 c = 15 1 15 3 5 x2 + 8x + 15 = x2 + 3x + 5x + 15 = x(x + 3) + 5(x + 3) = (x + 3) (x + 5) 5.4a S KM & PP

Factor with the Diamond: Example 7: x2 – 15x + 14 a = 1 b = -15 c = 14 1 14 2 7 x2 - 15x + 14 = x2 - 1x - 14x + 14 = x(x - 1) - 14(x - 1) = (x - 1) (x - 14) 5.4a S KM & PP

Factor with the Diamond: Example 9: x2 + 5x - 18 a = 1 b = 5 c = -18 1 18 2 9 3 6 There is no factor pair of 18 that will add or subtract to give 5 x2 + 5x - 18 cannot be factored using rational numbers 5.4a S KM & PP

Factor with the Diamond: Example 10: x2 - 36 a = 1 b = 0 c = -36 1 36 2 18 3 12 4 9 6 = x2 + 6x - 6x -36 = x(x+ 6) – 6(x +6) =(x+6)(x-6) Study this example carefully! The difference of squares is the product of conjugates. 5.4a S KM & PP

Factor with the Diamond: Example 11: 4x2 - 25 a = 4 b = 0 c = -25 1 100 2 50 4 25 5 20 10 = 4x2 + 10x - 10x -100 = 2x(2x + 5) – 5(2x +5) = (2x+5)(2x-5) Study this example carefully! The difference of squares is the product of conjugates. 5.4a S KM & PP

Factor with the Diamond: Example 12: 4x2 + 25 a = 4 b = 0 c = 25 1 100 2 50 4 25 5 20 10 There are no factors that add to 0 and multiply to +100 4x2 + 25 Study this example carefully! The sum of squares cannot be factored using rational numbers. 5.4a S KM & PP

Factor with the Diamond: Example 13: x2 -12x + 36 a = 1 b = -12 c = 36 1 36 2 18 3 12 4 9 6 = x2 - 6x - 6x +36 = x(x- 6) – 6(x -6) =(x-6)(x-6) =(x-6)2 A Binomial Squared! 5.4a S KM & PP

Factor with the Diamond: Example 14: x2 -12x - 36 a = 1 b = -12 c = -36 1 36 2 18 3 12 4 9 6 There are no factors that add to -12 and multiply to -36 x2 - 12x - 36 This trinomial cannot be factored using rational numbers. 5.4a S KM & PP

Factor with the Diamond: Example 15: 4x2 + 20x + 25 a = 4 b = 20 c = 25 1 100 2 50 4 25 5 20 10 = 4x2 + 10x + 10x +100 = 2x(2x + 5) + 5(2x +5) = (2x+5)(2x+5) = (2x+5)2 A Binomial Squared! 5.4a S KM & PP

Example 16: Don’t Forget the GCF! Always factor out the GCF first! Rewrite using the FOIL numbers Factor by Grouping The trinomial is completely factored! 5.4a S KM & PP

That’s All for Now! 5.4a S KM & PP