Distributive Law Special Forms and Tables Factoring Distributive Law Special Forms and Tables

1. Distributive Law a(b + c) = ab + ac Example Factor 2x2 + 4x = 2(x2) + 2(2x) = 2(x2 + 2x) 2x(x + 2) 2x2 + 4x = 2x(x + 2) Each term has a common factor of 2. Factor out the 2. And use DL to place it out in front. Each term also has a common X. Use the DL to factor it out front. Since the terms have nothing else in common, you are done.

2. Special Forms x2 – a2 = (x – a)(x + a) x3 – a3 = (x – a)(x2 + ax + a2) x3 + a3 = (x + a)(x2 – ax + a2) These are the basic special forms that you must remember!

x2 – a2 = (x – a)(x + a) x2 – 4 = x2 – 22 = (x – 2)(x + 2) To find a, take the square root of 4. So, a = = 2. To find a, take the square root of 7. So, a = . Since it doesn’t reduce, is the number.

x2 – a2 = (x – a)(x + a) x2 – 28 To find a, take the square root of 28.

x2 – a2 = (x – a)(x + a) 28 2 14 2 7 To find the square root of 28. Do a factor tree on 28. 2 and 7 are prime.

x2 – a2 = (x – a)(x + a) 28 2 14 2 7 = 2 Now, take the square root by boxing 2 of the same number. Place the number in each box out in front with a power equal to the number of boxes. All leftovers remain under the radical sign.

x2 – a2 = (x – a)(x + a) x2 – 28 = (x – 2 )(x + 2 ) To find a, take the square root of 28. Now that we have a = 2 , we can write the answer using the special form.

x3 – a3 = (x – a)(x2 + ax + a2) x3 – 27 To find a, take the cube root of 27.

x3 – a3 = (x – a)(x2 + ax + a2) 27 3 9 3 3 To find the cube root of 27. Do a factor tree on 27. 3 is prime.

x3 – a3 = (x – a)(x2 + ax + a2) 27 3 9 3 3 = 3 Now, take the cube root by boxing 3 of the same number. Place the number in each box out in front with a power equal to the number of boxes. All leftovers remain under the radical sign.

x2 – a2 = (x – a)(x + a) x3 – 27 = (x – 3)(x2 + 3x + 32) = (x – 3)(x2 + 3x + 9) To find a, take the cube root of 27. Now that we have a = 3, we can write the answer using the special form.

x3 + a3 = (x + a)(x2 – ax + a2) x3 + 64 a = 4 x3 + 64 = (x + 4)(x2 – 4x + 16) To find a, take the cube root of 64 as before. Use the special form and note that the only difference is the sign in the middle of each part.

3. Tables To set up a table to factor a quadratic, you need to identify the factors of the constant term (last number) that add to give you the number in front of the x in the middle.

3. Tables Factor x2 – 3x + 2 What factors of 2 (the last number) will add up to give you -3 (the number in front of x)? a b a + b

3. Tables Factor x2 – 3x + 2 (2)(1) = 2 but, 2 + 1 = 3, not -3. a b a + b 2 1 3

3. Tables Factor x2 – 3x + 2 When you have the right size (3) but the wrong sign (+ instead of -), change the signs in the a and b columns. a b a + b 2 1 3 -2 -1 -3

3. Tables Factor x2 – 3x + 2 Now the numbers work! (-2)(-1) = 2 (-2) + (-1) = -3 a b a + b 2 1 3 -2 -1 -3

3. Tables x2 – 3x + 2 = (x – 2)(x – 1) a b a + b 2 1 3 -2 -1 -3

3. Tables x2 – 3x + 2 = (x – 2)(x – 1) The signs of the numbers go in the factors. a b a + b 2 1 3 -2 -1 -3

Factoring Use the Methods in the following Order Distributive Law a(b + c) = ab + ac Special Forms x2 – a2 = (x – a)(x + a) x3 – a3 = (x – a)(x2 + ax + a2) x3 + a3 = (x + a)(x2 – ax + a2) Tables a b a + b