1. Any combination of the prime factorization. Find the number that “GAZINTA” all the numbers. 6 goes into 12, 2 times and into 18, 3 times. The only number that divides into 2 and 3 is 1…the number on the left side is the GCF. GCF = 6 The product of the numbers on the left side is the GCF. GCF = 4 * 2 = 8 There are no common factors in the remaining factors of 4, 6, and 9. The GCF = 4 The 6 and the 8 can still be divided by 2.

2. From the previous examples. Leftovers Distributive Property. When variables are a GCF, it will always be to the smallest power.

3. Group the terms in half. Factor each side by GCF’s Since the 1 st two terms are subtracting, both ( )’s will have minus signs. In fact, both sets of ( )’s must be the same for this to factor. If not the same, prime. Factor the same binomials as a GCF.

4. Factor by grouping

5. There is another pattern. Find the product of the F & L terms and O & I terms. Make up any two binomials, with no GCF, and FOIL them. F.O.I.L. When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials. When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the binomials.

6. This pattern gives us the a, b, c rule for finding our factors.Number Sense! Factor the trinomials. Factor by Grouping Answer looks like. Big Answer looks like. Did you just notice that the numbers in the binomial answers are the same numbers that were our factors? This will always happen when a = 1! Answer looks like. Number Sense Rules. Odd + Even = Odd Even + Even = Even Odd + Odd = Even 2 is a factor every Even. (Odd)*(Odd) = Odd

7. Factor the trinomials. 5(-12) = -60 not 60! Prime even + odd odd odd + odd even (odd)*(odd) = odd even + even even (odd)(even)=even even + even even odd + even odd odd + even odd odd + odd even odd even + even even

8. Prime doesn’t happen too often, so make sure you check everything! 1(-8) = -8 not 8! Prime even + even even odd + odd even odd

9. These directions means more than one factoring…Watch for GCF! GCF of 5 GCF of x 2 GCF of -3x 6 GCF of -2 Not Prime…factored -2 out! GCF of 4 Can’t go any further because of the variables cubed. even + even even

10. F.O.I.L. When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials. When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the FACTORS not the binomials.

11. The author actually suggested guessing what the binomials are and FOILing them out to test if the middle term is correct. 8 tries to get the right answer!?!

12. Answer looks like. Refers to the middle term. ODD + EVEN = ODD odd even Since we have an odd + even, we need odd factors. Break the 10 and 12 down to odd factors. Isolate the odd factors and multiply all possible odd combinations. Not the factors Right factors It should still factor if we switch the 15x and -8x. I can see a pattern! When you look at the left side of each factoring by grouping, I see the two binomials in the answer! Do you see that? Say YES! What terms are generating these binomials? Look above each step. It is the leading term and the two factors! Can we all agree that we will always factor out at least an x as the GCF? Yep. Here is a shortcut. Always put the “a” in both binomials. Put in the factors. Take out GCF’s 25

13. Refers to the middle term. EVEN + EVEN = EVEN Answer looks like this using the new short cut. Use 8x twice. even Because a = 8 Since we have an even + even, we factor out a 2 from our factors. Break the 8 down to get factors of 2’s. Put a 2( ) in each blank as a factor because we know that the two factors are even. 2( ) Factor 2 out of the -14. The sum of the two red ( )’s must = (– 7). Since – 7 is odd. Isolate the odd factors and multiply all possible odd combinations. Right factors! Put them in the red ( )’s! 6 and -20 are the two factors that add up to -14. Place them in our answer. Now we know we are not finished because we used the 8 twice. We have to divide out the extra 8 by finding the GCF of each binomial. 24 ( – 7 )

14. EVEN RULE ( odd + odd ) Factor. Refers to the middle term. ODD + ODD = EVEN Answer looks like this using new short cut. Use 3x twice. odd odd Because both a & c are odd Since a and c as odd factors we have an odd + odd = -34. This is going to take some time because all the factors will be odd. Break 63 down. Isolate each odd factor, from smallest to largest, and then multiply all possible odd combinations to create more odd factors. Wrong factors!Right factors! -7 and -27 are the two factors that add up to -34. Place them in our answer. Now we know we are not finished because we used the 3 twice. We have to divide out the extra 3 by finding the GCF of each binomial. 3 NO GCF

15. These directions means more than one factoring…Watch for GCF! even Because c = -8 The 2 or 4 must be multiplied to the 3 3 NO GCF Remember this example 2 pages ago, where the author FOILed it out 8 times? Which way is easier? odd even The 5 and -6 doesn’t work, so try 3 and 10! -3 and 10 work. 5 NO GCF GCF of 2x. Because 11 is much bigger than 4 and 3, multiply 4 and 3 to get 12. 4 NO GCF odd even even Because a = 4 One of the 2’s must be multiplied to the 5. 2 and 10 22

16. Factor completely. Need to have x powers in descending order. odd even odd No possible factors. PRIME 10 NO GCF odd even odd 63 is a big value… factors must be far apart. 3 odd even odd 3*_____ 5*_____ 15*____ 40 24 8 2 4 even even even 2( ) 2 One of the 3’s must be mult. to the 2, 6 and 3 subtract to be the 3 in the ( )’s. 3*_____ 5*_____ 9* ____ 15*____ 45*____ 60

17. Factor completely. GCF of 2x 2. odd odd even 3*_____ 5*_____ 9*____ 75 45 25 GCF of -3. odd even odd 32 3*_____ 7*_____ 21*____ 56 24 8 NO GCF 6 6 even even even 2( ) 2 The two 3’s mult. together, 9 and 5 add up to be the -14 in the ( )’s.

18. It is important to know that x 2, x 4, x even, etc. are all perfect squares Per. Sqr. Per. Sqr. Let’s try a ( ) 2 We must test the middle term! It Factors!

19. Done! NOT Done! ERASE Done! 4 and 36 are perfect squares, but 4 is a GCF! Done!

20. HIDE inside other polynomials! GCF of 9, 1 st !

21. Factor completely. Another Diff. of Per. SQ! NOT Diff. of Per. SQ! PRIME GCF of 3, 1 st ! GCF of -1, 1 st ! Or put the 25y 2 in front. Even powers on the variables are still perfect squares. Divide the powers by 2 to take the square root.

22. Middle terms exist! BinomialTrinomial Same sign as given Opposite sign as given Always Plus

23. Both Diff. of Per. Sq. and Perfect Cubes 1 st 2 nd

24. GCF ( LEFTOVERS ) The number of terms in the leftovers determines which step we go to. 1 st 2 nd Remember these like to hide inside of other polynomials. Same sign as given Opposite sign as given Always Plus

25. Remember the sign rules for what your answer looks like. GCF LT GCF RT SAME GCF LT GCF RT 3 to 1 SPLIT Difference of Perfect Squares 1 to 3 SPLIT is the same concept, but watch for signs!

26. GCF of 5 Step 2 D.P.S. Step 2 D.P.S. Again GCF of 2x Step 4 F. by G. Step 2 D.P.S. GCF of 3 Step 4 F. by G. Step 2 D.P.S. & P.C. Step 2 D.P.S. Step 2 & P.C. twice

27. Factor completely. GCF of 7 Step 3 GCF of 3x 2 Step 3 3 to 1 SPLIT Step 4 F. by G. Difference of Per. Squares 1 to 3 SPLIT Step 4 F. by G. Difference of Per. Squares GCF of -1 first. Distribute the minus!

28. Remember the product of -8(10)(3)(5)(0)(7)(11) = 0. Solve each for x.

29. Solve the equations by factoring. We don’t have to list the same number twice, but just know that there were two answers that were the same value. Never divide by the variable! Set the equation = 0. No reason to work out the 2 nd binomial because the only difference will be the sign.

30. Solve the equations by factoring. 2 odd even odd 3 The factors have to differ by 1, so 2(7)=14 and 3(5)=15

31. Solve the equations by factoring. even even even 2( ) One of the 3’s must be isolated, 3 and 18 will subtract to be 15 in the ( )’s. 42

32. Solve the equations by factoring. Will need to FOIL and set = 0 odd even odd 2

33. The cutting board is a rectangle because of the reference to “long and wide.” Build a rectangle. We know that the Length is twice the Width. The Area formula is L * W and the Area equals 800. The dimensions are 40 cm by 20 cm.

34. Means that the numbers differ by 1. The First number is unknown, call it x. The Second number must be 1 bigger… x + 1 Multiply the two numbers and set = 156 Assuming the racing number must be positive, the first number is 12 and the consecutive second number is 13. Means that the numbers differ by 2. The First number is unknown call it x. The Second number must be 2 bigger… x + 2 There are two sets of answers! -22 and -20 20 and 22

35. Right triangles have a special relationship called The Pythagorean Theorem. The legs of the right triangle are the sides of the right angle, labeled a and b. The hypotenuse is the longest side and is labeled c. The other two sides are 9 ft and 12 ft.

36. There will be 210 micrograms in the bloodstream at 3 minutes and 7 minutes.

37. The minimum length of the cable is 125 ft. The two distances are 30 ft. and 40 ft.

38. A number is 6 less than its square. Find all such numbers.