Factoring the Sum and Difference of Two cubes

Factoring the Sum and Difference of Two cubes 3 a 3 b + a 3 – b 3

Count 1 1 1 • How long is the edge? • How many squares in the face? • How many blocks?

Count Edge Face Blocks 1 1 1 2 4 8

Count Edge Face Blocks 1 1 1 2 4 8 3 9 27

Count Edge n 1 Face Blocks n 2 n 3 1 1 2 4 8 3 4 5 9 16 25 27 64 125

Memorize the First 10 Perfect Cubes n 1 2 3 4 5 6 7 8 9 10 n 2 1 4 9 16 25 36 49 64 81 100 n 3 1 8 27 64 125 216 343 512 729 1000

Recall the Difference of Two Squares Formula 2 b – =(a + b)(a – b) 2 x – 9 =(x + 3)(x – 3) • There are similar formulas for the sum and difference of two cubes.

Multiply a Binomial by a Trinomial The Sum of Cubes

Difference of Cubes

Compare the Formulas The Sum of Cubes The Difference of Cubes They are just alike except for where they are different.

Using the Difference of Cubes 3 x -8 Recall 23 = 8 = (x - 2 2)(x + 2 x + 4)

Using the Sum of Cubes 3 y + 27 Recall 33 = 27 = (y + 2 3)(y – 3 y + 9)

Factor Out the Common Factor 3 xa + 2 x + 21 a + 14 = 3 xa + 2 x + 3(7)a + 2(7) = x(3 a + 2) + 7(3 a + 2) = (3 a + 2)(x +7) This is called factoring by grouping.

What is factoring by grouping? Factoring a common monomial from pairs of terms, then looking for a common binomial factor is called factor by grouping. When do I use factoring by grouping? *when the problem consists of 4 terms How will my answer look? *it will be the product of two binomials

Factor the expression Notice there are two terms Pull the common factor out of each term. Notice what is left in each term after factoring out the common factor. Notice what each term has in common.

Try this example:

Factor the polynomial Form two binomials with a + sign between them.

Try this example:

6 x 2 – 3 x – 4 x + 2 by grouping 2 6 x – 3 x – 4 x + 2 2 = (6 x – 3 x) + (– 4 x + 2) = 3 x(2 x – 1) + -2(2 x - 1) = (2 x – 1)(3 x – 2)

Homework WB pp 89 and 90 Book p. 78 #1 -27 0 dd, p. 79 #1 -27 odd

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