Factor Diamond Practice Problems

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1 Factor Diamond Practice Problems 1. x 2 + 5x x 2 +7x x 2 + 9x x 2 + 9x x 2 7x x 2 x x 2 + x y 2 x x + 6x x -2x 2

2 Factor Diamond Practice Problems Walkthrough 1. x 2 + 5x Put the 6 on the top and the 5 on the bottom (x + 3)(x + 2) Now that you have the equation find the number which add or subtracts to make a 0 Lets start with the first one, (x + 3) a -3 would make it 0, (-3 + 3) = 0 For (x + 2) it would be a -2, (-2 + 2) = 0 So your two answers are 2. x 2 +7x Put the 12 on top and the 7 on the bottom Now its (x + 3)(x + 4), make them 0 For (x + 3) it would be -3, (-3 + 3) = 0 For (x + 4) it would be -4 (-4 + 4) = 0 So your two answers would be x 2 + 9x + 8 Put the 8 on top and the 9 on bottom Now its (x + 1)(x + 8) For (x + 1) its -1, (-1 + 1) = 0 For (x + 8) its -8, (-8 + 8) = 0 So your two answers would be

3 4. x 2 + 9x +14 Put the 14 on top and the 9 on bottom Now its (x + 2)(x + 7) For (x + 2) its -2, (-2 + 2) = 0 For (x + 7) its -7, (-7 + 7) = 0 So your two answers are 5. 2x 2 7x 4 Here you will be using the borrowing method, so multiply the first and last numbers, so the equation turns into x 2 7x -8 Now plug it into the diamond, the -8 on top and the -7 on bottom - Now its (x - 8)(x + 1) But since you borrowed a 2 you have to give it back, so divide both numbers by 2 - For (x - 8) it would turn into (x - 8/2) - For (x + 1) it would turn into (x + ½) And reduce them both - 8/2 reduces to 4, and ½ can t reduce any further so the equations are (x 4)(x + ½) - To get rid of the denominator multiply by it, in this problem it is (x + ½) so when multiplied by 2, it turns into (2x + 1) - Our final two equations are (x 4)(2x + 1) and make them both 0 - For (x 4) it would be 4, (4 4) = 0 - For (2x + 1) it would be -½, because 2 times -½ = -1, and (-1 + 1) = 0 - So your two final answers are So your two answers are

4 6. 3x 2 x 4 Multiply the first and last numbers, so the equation turns into x 2 x -12 Now put it into the diamond, -12 on top and the -1 on bottom - Now its (x 4)(x + 3) But you have to give back the 3 so (x 4/3)(x + 3/3) - Simplify them and if they cant be simplified anymore multiply by the denominator So (x 4/3) cant be simplified so you multiply by 3 which turns the problem into (3x 4) -For (x + 3/3) simplifies to (x + 1) - For (3x 4) it would be 4/3, (4-4) = 0 - For (x + 1) it would be -1, (-1 + 1) = 0 - So your two answers are 7. 5x 2 + x -18 Multiply the first and last numbers, so the equation turns into x 2 + x - 90 Now put it into the diamond, -90 on top and 1 on bottom - Now its (x 9)(x + 10) But you have to give back the 5 so (x 9/5)(x + 10/5) So (x 9/5) cant be simplified so you multiply by 5 which turns the problem into (5x 9) -For (x + 10/5) simplifies to (x + 2) - For (5x 9) it would be 9/5, (9-9) = 0 - For (x + 2) it would be -2, (-2 + 2) = 0 -So your final two answers are

5 8. 2y 2 y 1 Multiply the first and last numbers, so the equation turns into y 2 - y - 2 Now put it into the diamond, -2 on top and -1 on bottom - Now its (x 2)(x + 1) But you have to give back the 2 so (x 2/2)(x + 1/2) So (x 1/2) cant be simplified so you multiply by 2 which turns the problem into (2x 1) -For (x + 2/2) simplifies to (x + 1) - For (2x + 1) it would be -1/2, (-1 + 1) = 0 - For (x - 1) it would be 1, (1 1) = 0 So your two solutions are x + 6x 2 First you have to put it in descending order which changed the equation to 6x 2-13x + 6 and use the borrowing method - This changed the problem to x 2-13x + 36 Now put it into the diamond, 36 on top and -13 on bottom - Now its (x 9)(x - 4) But you have to give back the 6 so (x 9/6)(x - 4/6) So (x 9/6) simplifies to 3/2 and it cant be simplified anymore so you multiply by 2 which turns the problem into (2x 3) -When simplifying (x - 4/6) turns to (x - 2/3) and now you multiply by 3. This turns the problem into (3x - 2) -For (2x - 3) it would be 3/2, (3 3) = 0 - For (3x - 2) it would be 2/3, (-2 + 2) = 0 The two final answers are

6 x -2x 2 First you have to put it in descending order which changed the equation to -2x 2 + x + 15 and use the borrowing method - This changed the problem to x 2 + x - 30 Now put it into the diamond, -30 on top and 1 on bottom - Now its (x 5)(x + 6) But you have to give back the 2 so (x 5/2)(x + 6/2) So (x 6/2) simplifies to 3 which turns the problem into (x 3) -(x 5/2) cannot be simplified anymore so you multiply it by 2 which turns it into (2x - 5). -For (x - 3) it would be 3, (3 3) = 0 - For (2x - 5) it would be 5/2, (5-5) = 0 The two final answers are

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