2 x 6x 6 remember this rule is the difference of

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2 = (x + 6)(x – 6) Remember this rule is the “difference” of squares, meaning “-”. If an example is written as x 2 + a, it is not a difference of squares because “+” is used rather than “-” therefore making it a prime polynomial
FACTORING A DIFFERENCE OF CUBES The special factoring strategy for finding a difference of cubes can be written as: a 3 – b 3 = (a - b)(a 2 + ab + b 2 ) A few things to take note of: 1 - (a - b)(a 2 + ab + b 2 ) is a (binomial factor) x (trinomial factor) 2 – The binomial factor has the difference of the cube roots of the terms 3 – The terms in the trinomial factor are all positive 4 – The terms in the binomial factor determine the trinomial factor
FACTORING A DIFFERENCE OF CUBES For example if we were to factor the following: 4m 3 – 32n 3 First we need to factor out the common factor. What number is common in both 4 and 32? That number is 4. Now, we can rewrite this as: 4(m 3 -8n 3 ) Now, we look at the 8 in the rewritten problem. We can break 8n 3 down further by asking what number cubed or multiplied by itself three times equals 8? This number is 2. (2 x 2 = 4 and 4 x 2 = 8 so 2 x 2 x 2 =8) Now, we can rewrite this as: 4[m 3 -(2n) 3 ] (Continued on next slide)
FACTORING A DIFFERENCE OF CUBES Before we go to the next step, let’s remember the formula for a difference of cubes… That is a 3 – b 3 = (a - b)(a 2 + ab + b 2 ) So now that we have the problem written as 4[m 3 -(2n) 3 ] we can use the formula to break it down further. We will let a = m and 2n = b We rewrite the problem again using the formula (a - b)(a 2 + ab + b 2 ): 4(m - 2n)[m 2 + m(2n) + (2n) 2 ] Apply any exponents and multiply. Finally, rewrite: 4(m - 2n)(m 2 + 2mn + 2n 2 )

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